The probability seminar is organized by Vivian Healey, Steven Lalley, Gregory Lawler and Xinyi Li. It takes place on Fridays at 2:30 pm in Eckhart 202, unless otherwise specified.

*Title:* ** Multiple SLEs, discrete interfaces, and crossing probabilities
**

*Abstract: Multiple SLEs are conformally invariant measures on families of curves, that naturally correspond to scaling limits of interfaces in critical planar lattice models with alternating (”generalized Dobrushin”) boundary conditions. I discuss classification of these measures and how the convergence for discrete interfaces in many models is obtained as a consequence. When viewed as measures with total mass, the multiple SLEs can also be related to probabilities of crossing events in lattice models.
The talk is based on joint works with
Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)
and Vincent Beffara (Université Grenoble Alpes, Institut Fourier). *

*Title:* ** On the range of lattice models in high dimensions.**

*Abstract:* We investigate the scaling limit of the {\em range} (the set
of visited vertices) for a general class of critical lattice models,
starting from a single initial particle at the origin. Conditions
are given on the random sets and an associated ``ancestral relation''
under which, conditional on long term survival, the rescaled ranges
converge weakly to the range of super-Brownian motion as random
sets. These hypotheses also give precise asymptotics for the limiting
behaviour of the probability of exiting a large ball. Applications
include voter models, contact processes, oriented percolation and lattice
trees. This is joint work with Mark Holmes and also features work
of Akira Sakai and Gordon Slade.

*Title:* ** A Tale of Two Integrals**

*Abstract:* I will discuss the asymptotic behavior of two matrix integrals: the Harish-Chandra/Itzykson-Zuber integral, and its additive counterpart, the Brezin-Gross-Witten integral. Both are integrals over the group U(N), and their behavior in the large N limit is the subject of a pair of conjectures formulated by physicists in 1980 in connection with the large N limit in lattice gauge theory. I will discuss a proof of these conjectures which involves relating them to fundamental combinatorial structures, in particular Hurwitz numbers and increasing subsequences in permutations.

*Title:* ** Scaling limit of uniform spanning tree in three dimensions**

*Abstract:* In the talk, we will show the existence of the scaling limit of three-dimensional uniform spanning tree (UST) with respect to the Gromov-Hausdorff-Prohorov type topology and obtain several properties of the limiting tree. Moreover, we will prove that the rescaled simple random walk on the 3D UST converges weakly to a diffusion on the limit tree above. Detailed transition density estimates for the limiting process will be derived. These are ongoing works with O. Angel (UBC), D. Croydon (Kyoto University) and S. Hernandez Torres (UBC).

*Title:* ** Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential**

*Abstract:* I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

*Title:* ** Laplacian growth and sandpiles on the Sierpinski gasket: limit shape universality, fluctuations, and beyond **

*Abstract:* Given a locally finite connected graph with a distinguished vertex $o$, start $m$ particles at $o$, and let them aggregate according to one of the four following discrete Laplacian growth models: internal diffusion-limited aggregation (IDLA), rotor-router aggregation (RRA), divisible sandpiles, and abelian sandpiles. We are interested in describing the limit shapes and radial growth of the cluster in each model. Do they coincide for all 4 models? And how sharp a radial bound can one get?

On the Euclidean lattice: No, though there are many results on radial bounds. On the Sierpinski gasket graph, where $o$ is the corner vertex: Yes, and sharp radial bounds are now available.

My talk will address the gasket story and consists of two parts:

1) By solving the divisible sandpile problem, one can gain access to information about the harmonic measure and obtain limit shape results for IDLA and RRA. I will describe how the proofs work on the gasket, incorporating ideas from the earliest IDLA proofs (Lawler-Bramson-Griffeath), random walks on graphs (Harnack inequality), and a fast simulation algorithm (Friedrich-Levine).

2) The abelian sandpile problem poses an entirely different set of challenges. Quite luckily, on the gasket we have solved the abelian sandpile growth problem EXACTLY. The key idea is to exploit the cellular structure, the cut point structure, and axial symmetries to perform systematic topplings in waves. This allows us to inductively establish tiling patterns in the sandpile configurations, in particular, the identity elements of the associated sandpile groups. It also leads to the enumeration of all the radial jumps in the growing cluster, and implies, via the renewal theorem, a radial asymptotic formula in the form of a power law modulated by log-periodic oscillations---the best possible result on such a state space.

Based on joint works with Wilfried Huss, Ecaterina Sava-Huss (TU Graz), Alexander Teplyaev (UConn), and Jonah Kudler-Flam (UChicago).

*Title:* ** Dynamical percolation on random triangular lattices**

*Abstract:*
Dynamical (site) percolation on a graph is a Markov process where the state space is the set of possible black/white colorings of the vertices of the graph. Each vertex is associated with an independent Poisson clock, and the color of a vertex is resampled every time its clock rings. Dynamical percolation on the regular triangular lattice was thoroughly studied by Garban, Pete and Schramm. In this talk we will discuss the case when the graph is a uniformly sampled triangulation. In particular, we will explain how to describe the scaling limit of this process and show its ergodicity. Time permitting, we will also explain the role it played in the study of the conformal structure of uniform triangulations. Based on joint work with Christophe Garban, Nina Holden and Avelio Sepulveda.

*Title:* ** Localization phenomena of directed polymers**

*Abstract:* On the d-dimensional integer lattice, directed polymers are paths of a random walk that have been reweighted according to a random environment that refreshes at each time step. The qualitative behavior of the system is governed by a temperature parameter; if this parameter is small, the environment has little effect, meaning all possible paths are close to equally likely. If the parameter is made large, however, the system undergoes a phase transition beyond which the path ''localizes,'' meaning the polymer measure concentrates. In this talk, I will discuss different quantitative statements of this phenomenon, methods of proving these statements, and natural connections to other statistical mechanical systems. (Joint work with Sourav Chatterjee)

*Title:* ** The effect of random modifications on Loewner hulls**

*Abstract:* Loewner hulls are determined by their real-valued driving functions, via the Loewner differential equation. We will discuss two projects which study the geometric effect on the Loewner hulls of random modifications. In the first project, which is joint work with Kei Kobayashi and Andrew Starnes, the driving function is composed with a random time change, such as the inverse of an $\alpha$-stable subordinator. In contrast to Schramm-Loewner evolution (SLE), we show that for a large class of random time changes, the time-changed Brownian motion process does not generate a simple curve. Further we develop criteria which can be applied in many situations to determine whether the Loewner hull generated by a time-changed driving function is simple or non-simple. In the second project, joint with Nathan Albin and Pietro Poggi-Corradini, we look at the scaling limit of Peano curves associated to so-called fair spanning trees. In contrast to the convergence of the uniform spanning tree Peano curve to SLE(8), we show that the scaling limit is a deterministic object.

*Title:* ** A geometric property for optimal paths and its applications in first passage percolation**

*Abstract:* We consider the first passage percolation model in Z^d with a weight distribution F for 0 < F(0) < p_c. In this paper, we derive a geometric property for optimal paths to show that all of them have to pass an M-exit. By this property, we show that the shape is strictly convex, and we solve the height problem.

*Title:* **Unique continuation and localization on the planar lattice **

*Abstract:* I will discuss joint work with Jian Ding in which we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain-Kenig and uses a new unique continuation result inspired by Buhovsky-Logunov-Malinnikova-Sodin.

*Title:* ** Universality and delocalization of band random matrix**

*Abstract:* In this talk, we discuss some recent work related to a main conjecture on random matrix theory, i.e. phase-transition conjecture on random matrix theory.

The prediction says that phase-transition occurs at the band width regime W \sim N^{1/2}. For high dimensional matrix, i.e. x,y\in Z^d, H_{xy}, there exists some similar stimulation results.

Based on the development of studying on resolvent, i.e., G=(H-z)^{-1}, we obtained some results on low and high dimension cases. In this talk, we will introduce these work and the main ideas and tools used in these work.

They are jointed work with Laci Erdos, Paul Bourgade, H.T. Yau, Yang Fan, etc.

*Title:* **A support theorem for SLE curves**

*Abstract:* SLE curves are an important family of random curves in the plane. They share many similarities with solutions of SDE (in particular, with Brownian motion). Any question asked for the latter can be asked for the former. Inspired by that, Yizheng and I investigate the support for SLE curves. In this talk, I will explain our theorem with more motivation and idea.

*Title:* **Large concentration of harmonic measure: discrete and continuous case. **

*Abstract:* I will discuss the sharp bounds on the multifractal spectrum of planar harmonic measure and their continuous analogues. In particular, I will talk about the refinements of Beurling estimates on the concentration of harmonic measure.

*Title:* **Invertibility of adjacency matrices for random d-regular graphs **

*Abstract:* The singularity problem of random matrices asks the probability that a given discrete random matrix is singular. The first such result was obtained by Komlós in 1967. He showed a Bernoulli random matrix is singular with probability o(1). This question can be reformulated for the adjacency matrices of random graphs, either directed or undirected. The most challenging case is when the random graph is sparse. In this talk, I will prove that for random directed and undirected d-regular graphs, their adjacency matrices are invertible with high probability for all d>=3. The idea is to study the adjacency matrices over a finite field, and the proof combines a local central limit theorem and a large deviation estimate.

*Title:* ** The directed landscape**

*Abstract:* The longest increasing subsequence in a random permutation, the second
class particle in TASEP, and semi-discrete polymers at zero temperature
have the same scaling limit: a random function with Holder exponent
2/3-. This limit can be described in terms of the directed landscape, a
random metric at the heart of the Kardar-Parisi-Zhang universality class.
Joint work with Duncan Dauvergne and Janosch Ortmann.

*Title:* **Chemical distance for level-set percolation of planar metric-graph Gaussian free field **

*Abstract:* We consider percolation of the level-sets of a metric-graph Gaussian free field on a box of the planar integer lattice. We provide an upper bound for the length of the shortest path joining the boundary components of a macroscopic annulus. The bound holds with high probability conditioned on connectivity and is sharp up to a poly-logarithmic factor with exponent one-quarter. This is joint work with Jian Ding.

*Title:* ** Bismut's Interpolation Between Riemannian Brownian Motion and Geodesic Flow**

*Abstract:* A few years ago Bismut found a natural family of diffusion processes that interpolates between Brownian motion and the geodesic flow
on a compact Riemannian manifold. The convergence to Brownian motion (together with an attendant Gaussian field) at one end
of the parameter interval is nontrivial and proved in the weak sense of finite dimensional marginal distributions. By slightly extending the traditional
setting for stochastic calculus on manifold we show that the convergence can be realized as a strong one in the path space. Such a more precise
convergence to Brownian motion will help us in understanding asymptotic behavior of some classical functional inequalities for path spaces.

*Title:* ** Random walks with local memory**

*Abstract:* The theme of this talk is walks in a random environment of ``signposts'' altered by the walker. I'll focus on three related examples:

1. Rotor walk on Z^2. Your initial signposts are independent with the uniform distribution on {North,East,South,West}. At each step you rotate the signpost at your current location clockwise 90 degrees and then follow it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps you will visit order n^{2/3} distinct sites. I'll outline an elementary proof of a lower bound of this order. The upper bound, which is still open, is related to a famous question about the path of a light ray in a grid of randomly oriented mirrors. This part is joint work with Laura Florescu and Yuval Peres.

2. p-rotor walk on Z. In this walk you flip the signpost at your current location with probability 1-p and then follow it. I'll explain why your scaling limit will be a Brownian motion perturbed at its extrema. This part is joint work with Wilfried Huss and Ecaterina Sava-Huss.

3. p-rotor walk on Z^2. Rotate the signpost at your current location clockwise with probability p and counterclockwise with probability 1-p, and then follow it. This walk ``organizes'' its environment by destroying cycles of signposts. A native environment -- stationary in time, from your perspective as the walker -- is an orientation of the uniform spanning forest, plus one additional edge. This part is joint work with Swee Hong Chan, Lila Greco, and Peter Li.

*Title:* **2-RSB for spherical mixed p-spin models at zero temperature**

*Abstract:* Historically, the method of replica symmetry breaking (RSB) was introduced by Parisi to study mean field spin glass models. This method has played indispensable role in the physical deduction of Parisi formula. Mathematically, the level of RSB corresponds
to the number of points in the support of the Parisi measure. There have been known Replica Symmetric, 1-step RSB and Full-step RSB 'natural' spin glass models. However, k-step RSB for finite k>1 is less well understood. In this talk, I will show that certain
spherical mixed p-spin glass models are 2-step RSB at zero temperature and present consequences in the energy landscape.
This talk is based on joint work with Antonio Auffinger.

*Title:* ** Natural parametrization for the scaling limit of loop-erased random walk in three dimensions**

*Abstract:* We will consider loop-erased random walk (LERW) and its scaling limit in three dimensions. Gady Kozma (2007) shows that as the lattice spacing becomes finer, LERW in three dimensions converges weakly to a random compact set with respect to the Hausdorff distance. We will show that 3D LERW parametrized by renormalized length converges in the lattice size scaling limit to Kozma's scaling limit parametrized by some suitable measure on it with respect to the uniform norm. This is based on joint works with Xinyi Li (University of Chicago).

*Title:* ** Percolation on random trees** (joint work with
Marcus Michelen and Josh Rosenberg)

*Abstract:* Let T be a tree chosen from Galton-Watson measure and
let {U_v} be IID uniform [0,1] random variables associated
with the edges between each vertex and its parent.
These define coupled Bernoulli percolation processes,
as well as an invasion percolation process.
We study quenched properties of these percolations:
properties conditional on T that hold for almost every T.
The invasion cluster has a backbone decomposition
which is Markovian if you put on the right blinders.
Under suitable moment conditions, the law of the
(a.s. unique) backbone ray is absolutely continuous
with respect to limit uniform measure.
The quenched survival probabilities are smooth in the
supercritical region p > p_c. Their behavior as p -> p_c
depends on moment assumptions for the offspring distribution.

*Title:* ** Quantitative CLTs for random walks in random environments**

*Abstract:*
The Berry-Esseen estimates give quantitative error estimates on the CLT for sums of i.i.d. random variables, and the polynomial decay rate for the error depends on moment bounds of the i.i.d. random variables with the optimal $1/\sqrt{n}$ rate of convergence obtained under a third moment assumption. In this talk we will prove quantitative error bounds for CLTs of random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.

*Title:* **Algorithmic Pirogov-Sinai theory**

*Abstract:* We develop efficient algorithms to approximate the partition function and sample from the hard-core and Potts models on lattices at sufficiently low temperatures in the phase coexistence regime. In contrast, the Glauber dynamics are known to take exponential time to mix in this regime. Our algorithms are based on the cluster expansion and Pirogov-Sinai theory, classical tools from statistical physics for understanding phase transitions, as well as Barvinok's approach to polynomial approximation. Joint work with Tyler Helmuth and Guus Regts.

*Title:* **Yaglom-type limit theorems for branching Brownian motion with absorption**

*Abstract:* We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, improving upon a result of Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, which also improves upon results of Kesten (1978). An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.

* Title:* **Phase transitions of random constraint satisfaction problems **

*Abstract:* Random constraint satisfaction problems encode many interesting questions in the study of random graphs such as the chromatic and independence numbers. Ideas from statistical physics provide a detailed description of phase transitions and properties of these models. We will discuss the one step replica symmetry breaking transition that many such models undergo.

* Title:* **Coalescence of polymers in Last Passage Percolation **

*Abstract:*
We will discuss new bounds on the time for polymers in first passage percolation to coalesce. As a consequence we will prove that there are no non-trivial bi-geodesics. The methods will combine bounds given by exactly solvable calculations with tools from percolation theory.

*Title:* **Stability of the elliptic Harnack Inequality**

*Abstract:* Following the work of Moser, as well as de Giorgi and Nash,
Harnack inequalities have proved to be a powerful tool in PDE as well as in
probability. In the early 1990s Grigor'yan and Saloff-Coste
gave a characterisation of the parabolic Harnack inequality (PHI).
This characterisation implies that the PHI is stable under bounded perturbation
of weights, as well as rough isometries. In this talk we prove
the stability of the EHI. The proof uses the concept of a quasi symmetric
transformation of a metric space, and the introduction of these ideas to
Markov processes suggests a number of new problems.
This is joint work with Mathav Murugan (UBC).

*Title:* **Factorizations and estimates of Dirichlet heat kernels for non-local operators with critical killings **

*Abstract:* In this talk I will discuss heat kernel estimates for critical perturbations
of non-local operators. To be more precise, let $X$ be the reflected
$\alpha$-stable process in the closure of a smooth open set $D$, and
$X^D$ the process killed upon exiting $D$. We consider potentials of the
form $\kappa(x)=C\delta_D(x)^{-\alpha}$ with positive $C$ and the
corresponding Feynman-Kac semigroups. Such potentials do not belong
to the Kato class. We obtain sharp two-sided estimates for the heat
kernel of the perturbed semigroups. The interior estimates of the
heat kernels have the usual $\alpha$-stable form, while the boundary
decay is of the form $\delta_D(x)^p$ with non-negative
$p\in [\alpha-1, \alpha)$ depending on the precise value of the
constant $C$. Our result recovers the heat kernel estimates of both
the censored and the killed stable process in $D$. Analogous
estimates are obtained for the heat kernel of the Feynman-Kac
semigroup of the $\alpha$-stable process in
${\mathbf R}^d\setminus \{0\}$ through the potential $C|x|^{-\alpha}$.
All estimates are derived from a more general result described as follows:
Let $X$ be a Hunt process on a locally compact separable metric space in
a strong duality with $\widehat{X}$. Assume that transition densities of
$X$ and $\widehat{X}$ are comparable to the function $\widetilde{q}(t,x,y)$
defined in terms of the volume of balls and a certain scaling function.
For an open set $D$ consider the killed process $X^D$, and a critical
smooth measure on $D$ with the corresponding positive additive functional
$(A_t)$. We show that the heat kernel of the the Feynman-Kac semigroup
of $X^D$ through the multiplicative functional $\exp(-A_t)$ admits the
factorization of the form
${\mathbf P}_x(\zeta >t)\widehat{\mathbf P}_y(\widehat{\zeta}>t)\widetilde{q}(t,x,y)$.
This is joint work with Soobin Cho, Panki Kim and Zoran Vondracek.

*Title:* ** Ergodic theory of the stochastic Burgers equation**

*Abstract:* The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.

*Title:* **Local density estimate for a hypoelliptic SDE **

*Abstract:* In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting.
The talk is based on an ongoing project with Xi Geng and Samy Tindel.

*Title:* ** The weight, geometry and coalescence of scaled polymers in Brownian last passage percolation**

*Abstract:* In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice consisting of independent and identically distributed weights is considered. The weight of an upright path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation.
We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression. Building on a study arXiv:1609.02971
concerning the decay in probability for the existence of several near polymers with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in scaled coordinates has a superpolynomial decay rate in k.
This result has implications for the Brownian regularity of the scaled polymer weight profile begun from rather general initial data.

*Title:* ** Stationary Harmonic Measure and DLA in the Upper Half Plane**

*Abstract:* In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.

*Title:* ** Geodesics in First-Passage Percolation **

*Abstract:* First-passage percolation is a classical random growth model which
comes from statistical physics. We will discuss recent results about
the relationship between the limiting shape in first passage
percolation and the structure of the infinite geodesics. This includes
a solution to the midpoint problem of Benjamini, Kalai and Schramm.
This is joint work with Daniel Ahlberg.

*Title:* ** Shifted weights and restricted path length in first-passage percolation
**

*Abstract:* First-passage percolation has remained a challenging field of study since its introduction in 1965 by Hammersley and Welsh. There are many outstanding open problems. Among these are properties of the limit shape and the Euclidean length of geodesics. This talk describes a convex duality between a shift of the edge weights and the length of the geodesic, together with related results on the regularity of the limit shape as a function of the shift. The talk is based on joint work with Arjun Krishnan (Rochester) and Firas Rassoul-Agha (Utah).

*Title:* ** The giant component in a degree-bounded process**

*Abstract:* Graph processes $(G(i),i\ge 0)$ are usually defined as follows. Starting from the empty graph on $n$ vertices, at each step $i$ a random edge is added from a set of available edges. For the $d$-process, edges are chosen uniformly at random among all edges joining vertices of current degree at most $d-1$.
The fact that, during the process, vertices become 'inactive' when reaching degree $d$ makes the process depend heavily on its history. However, it shares several qualitative properties with the classical Erdos-Renyi graph process. For example, there exists a critical time $t_c$ at which a giant component emerges, whp (that is, the largest component in $G(tn)$ goes from logarithmic to linear order).
In this talk we consider $d\ge 3$ fixed and describe the growth of the size of the giant component. In particular, we show that whp the largest component in $G((t_c+\eps)n)$ has asymptotic size $cn$, where $c\sim c_d \eps$ is a function of time $\eps$ as $\eps \to 0+$.
The growth, linear in $\eps$, is a new common qualitative feature shared with the Erdos-Renyi graph process and can be generalized to hypergraph processes with different max-allowed degree sequences. This is work in progress jointly with Lutz Warnke.

*Title:* ** Phase transitions in the 1-2 model**

*Abstract:* A configuration in the 1-2 model is a subgraph of the hexagonal lattice, in which each vertex is incident to 1 or 2 edges. By assigning weights to configurations at each vertex, we can define a family of probability measures on the space of these configurations, such that the probability of a configuration is proportional to the product of weights of configurations at vertices. We study the phase transition of the model by investigating the probability measures with varying weights. We explicitly identify the critical weights, in the sense that the edge-edge correlation decays to 0 exponentially in the subcritical case, and converges to a non-zero constant in the supercritical case, under the limit measure obtained from torus approximation. These results are obtained by a novel measure-preserving correspondence between configurations in the 1-2 model and perfect matchings on a decorated graph, which appears to be a more efficient way to solve the model, compared to the holographic algorithm used by computer scientists to study the model. The major difficulty here is the absence of stochastic monotonicity.

*Title:* ** Fluctuations of the free energy of spherical Sherrington-Kirkpatrick model
**

*Abstract:* Consider the question of finding the maximum of a random polynomial defined on a closed manifold or a finite graph. Spherical Sherrington-Kirkpatrick (SSK) model is a finite temperature version of this question when the underlying space is a sphere. The free energy is the finite temperature version of the maximum value. The limit of the free energy as the dimension of the sphere becomes infinity is known by the works of Parisi, Cristanti, Sommers, and Talagrand. In this talk we consider the fluctuations when the polynomial is a symmetric quadratic function. We use a connection to random matrices and obtain limit theorems. This is a joint work with Ji Oon Li and Hao Wu.

*Title:* **Conformally Invariant Paths and Loops**

*Abstract:*
Abstract: There has been incredible progress in the last twenty years in the rigorous understanding of two-dimensional critical systems in statistical physics. I will give an overview with an emphasis on several related models: loop-erased random walk, spanning trees and corresponding loop soup; fractal paths and loops arising in critical systems (Schramm-Loewner evolution); the Gaussian free field and functions thereof (quantum gravity). I will also discuss some challenges for the future. This talk is intended for a general audience - it is not assumed that the audience is familiar with these terms.

*Title:* ** Phase transitions in some percolation models with long-range correlations on general graphs**

*Abstract:*
Abstract: We consider two fundamental percolation models with long-range correlations on reasonably general and well-behaved transient graphs: The Gaussian free field and (the vacant set) of Random Interlacements. Both models have been the subject of intensive research during the last years and decades, on $\Z^d$ as well as on some more general graphs. We consider their percolation phase transition and investigate a couple of interesting properties of their critical parameters, in particular the existence of a phase transition.
This talk is based on joint works with A. Prevost (Koeln) and P.-F. Rodriguez (Los Angeles).

*Title:* ** Understanding rare events in models of statistical mechanics**

*Abstract:* Statistical mechanics models are ubiquitous at the interface of probability theory, information theory, and inference problems in high dimensions. In this talk we will focus on sparse
graphs, and polymers on lattices; two canonical models in natural sciences. The study of large deviations is intimately related to the understanding of such models. We will consider the rare events that a sparse random network has an atypical number of certain local
structures and that a polymer in random media has atypical weight. Conditioning on such events can produce different, ranging from local to more global, geometric effects. We will discuss some such results obtained, relying on a variety of entropy theoretic, combinatorial, and analytic tools.

*Title:* ** Stationary Harmonic Measure and DLA in the Upper Half Plane**

*Abstract:* In this talk, we introduce the stationary harmonic measure in the upper half plane. By bounding this measure, we are able to define both the discrete and continuous time diffusion limit aggregation (DLA) in the upper half plane with absorbing boundary conditions. We prove that for the continuous model the growth rate is bounded from above by $o(2+\epsilon)$. When time is discrete, we also prove a better upper bound of $o(2/3+\epsilon)$, on the maximum height of the aggregate at time $n$.

*Title:* **Relating a classical planar map embedding algorithm to Liouville quantum gravity and SLE(16)
**

*Abstract:* In 1990, Walter Schnyder introduced a class of 3-spanning-tree decompositions of a simple triangulation to describe a combinatorially natural grid embedding algorithm for planar maps. It turns out that a uniformly sampled Schnyder-wood-decorated triangulation on n vertices converges as n tends to infinity to a random fractal surface, called a Liouville quantum gravity (LQG), together with a triple of intertwined fractal curves known as SLE(16). We will motivate this result by describing Schnyder’s algorithm and discussing some history of random planar map convergence results, and we will also introduce LQG and SLE and explain their role in the story.

*Title:* ** New perspectives on Mallows permutations
**

*Abstract:* I will discuss two projects concerning Mallows permutations, with Ander Holroyd, Tom Hutchcroft and Avi Levy. First, we relate the Mallows permutation to stable matchings, and percolation on bipartite graphs.
Second, we study the scaling limit of the cycles in the Mallows permutation, and relate it to diffusions and continuous trees.

*Title:* ** Dynamical freezing in a spin glass system with logarithmic correlations**

*Abstract:* We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. Joint work with Aser Cortines (University of Zurich) and Oren Louidor (Technion).

*Title:* ** The ant in high dimensional labyrinths**

*Abstract:* One of the most famous open problem in random walks in random environments is to understand the behaviour of a simple random walk on a critical percolation cluster, a model known as the ant in the labyrinth. We will present new results on the scaling limit for the simple random walk on the critical branching random walk in high dimension which converges, after scaling, to the Brownian motion on the integrated super-brownian motion. In the light of lace expansion, we believe that the limiting behaviour of this model should be universal for simple random walks on critical structures in high dimension. In particular, recent progress show
that similar results hold for lattice trees.

*Title:* ** Stationary aggregation processes**

*Abstract:* In this talk I'll introduce stationary versions of known aggregation models e.g., DLA, Hastings Levitov, IDLA and Eden. Using the additional symmetry and ergodic theory, one obtains new geometric insight on the aggregation processes.

* Title:* ** Gravitational allocation to uniform points on the sphere**

*Abstract:* Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition-with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). Joint work with Nina Holden and Alex Zhai.

* Title:* ** The strange geometry of high-dimensional random spanning forests.**

*Abstract:* The uniform spanning forest (USF) in the lattice Z^d, first studied by Pemantle (Ann. Prob. 1991), is defined as a limit of uniform spanning trees in growing finite boxes. Although the USF is a limit of trees, it might not be connected- Indeed, Pemantle proved that the USF in Z^d is connected if and only if d<5. In later work with Benjamini, Kesten, and Schramm (Ann. Math 2004) we extended this result, and showed that the component structure of the USF undergoes a phase transition every 4 dimensions: For dimensions d between 5 and 8 there are infinitely many trees, but any two trees are adjacent; for d between 9 and 12 this fails, but for every two trees in the USF there is an intermediary tree, adjacent to each of the them. And this pattern continues, with the number of intermediary trees required increasing by 1 every 4 dimensions. In this talk, I will show that this is not the whole story, and for d>8 the USF geometry undergoes a qualitative change every time the dimension increases by 1. (Joint work with Tom Hutchcroft.)

*Title:* ** Random matrices, the Riemann zeta function and trees**

*Abstract:* Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to predict the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on joint works with Arguin, Belius, Radziwill and Soundararajan.

*Title:* ** Local limits of Random Sorting Networks**

*Abstract:* A sorting network is a shortest path between 12..n and n..21 in the Cayley graph of the symmetric group spanned by swaps of adjacent letters. We will discuss the bulk
local limit of the swap process of uniformly random sorting networks and encounter
universal distributions of the random matrix theory, including the celebrated
Gaudin-Mehta law, which describes the energy level spacings in heavy nuclei.

*Title:* ** Geodesics in First-Passage Percolation **

*Abstract:* First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This incudes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.

*Title:* ** Regenerative permutations: Mallows(q) and Riemann zeta function**

*Abstract:* In this talk we discuss regenerative permutations on integers, with emphasis on two particular models: p-shifted and P-biased permutations. When p is the geometric distribution, the p-shifted permutations appear to be the limit of Mallows permutation model. We generalize and simplify previous work of Gnedin and Olshanski. The P-biased permutations are reminiscent of successive sampling in Bayesian statistics. Interestingly, some zeta formulas appear in the evaluation of renewal quantities of GEM-biased permutations. This is based on joint work with Jean-Jil Duchamps and Jim Pitman.

*Title:* ** Extremal metrics, eigenvalues, and graph separators**

*Title:* ** Discrete conformal metrics and spectral geometry on distributional limits**

*Title:* ** The SK model is FRSB at zero temperature**

*Abstract:* In the early 80's, the physicist Giorgio Parisi wrote a series of ground breaking papers where he introduced the notion of replica symmetry breaking. His powerful insight allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times.
In this talk, we will prove Parisi's prediction at zero temperature for the mixed p-spin model, a generalization of the SK model. We will show that at zero temperature the functional order parameter is full-step replica symmetry breaking (FRSB). We will also describe the importance of this result for the description of the energy landscape.
Based on recent works with Wei-Kuo Chen (U. of Minnesota) and Qiang Zeng (Northwestern U.).

*Title:* ** Limit theory for statistics of random geometric structures**

*Abstract:* Questions arising in stochastic geometry and applied geometric probability are often understood in terms of the behavior of statistics of large random geometric structures.Such structures arise in diverse settings and include:
(i) Point processes of dependent points in R^d, including determinantal, permanental, and Gibbsian point sets, as well as the zeros of Gaussian analytic functions,
(ii) Simplicial complexes in topological data analysis,
(iii) Graphs on random vertex sets in Euclidean space,
(iv) Random polytopes generated by random data.
Global features of geometric structures are often expressible as a sum of local contributions. In general the local contributions have short range spatial interactions but complicated long range dependence. In this survey talk we review ``stabilization'' methods for establishing the limit theory for statistics of geometric structures. Stabilization provides conditions under which the behavior of a sum of local contributions is similar to that of a sum of independent identically distributed random variables.

*Title:* ** The Structure of Extreme Level Sets in Branching Brownian Motion**

*Abstract:* We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida (joint work with A. Cortines, O. Louidor).

*Title:* ** Branching capacity and critical branching random walks**

*Abstract:* In this talk, I will introduce branching capacity for any finite subset of Z^d (d>=5). It turns out to be an important subject in the study of critical branching random walks. I will discuss its connections with critical branching random walks from the following three perspectives: 1) the hitting probability of a set by critical branching random walk; 2) branching recurrence and branching transience; 3) the local limit of critical branching random walk in torus.

*Title:* ** SLE loop measures**

*Abstract:* An SLE loop measure is a $\sigma$-finite measure on the space of loops, which locally looks like a Schramm-Loewner evolution (SLE) curve. In this work, we use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb C}$, which satisfy M\"obius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parameterized by its $(1+\frac \kappa 8)$-dimensional Minkowski content. Second, by integrating rooted SLE$_\kappa$ loop measures, we construct the unrooted SLE$_\kappa$ loop measure in $\widehat{\mathbb C}$, which satisfies M\"obius invariance and reversibility. Third, we extend the SLE$_\kappa$ loop measures from $\widehat{\mathbb C}$ to subdomains of $\widehat{\mathbb C}$ and to Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLE$_\kappa$ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLE$_\kappa$ loop measures for $\kappa\in(0,4]$ give examples of Malliavin-Kontsevich-Suhov loop measures for all $c\le 1$. The space-time homogeneity of rooted SLE$_\kappa$ loop measures in $\widehat{\mathbb C}$ answers a question raised by Greg Lawler.

*Title:* ** Extremal and local statistics for gradient field models**

*Abstract:* We study the gradient field models with uniformly convex potential (also known as the Ginzburg-Landau field) in two dimension. This is a log-correlated (but generally non-Gaussian) random field that arises in different branches of mathematical physics. Previous results (Naddaf-Spencer, and Miller) were focused on the CLT for the linear functionals of the field. In this talk I will describe more precise results on the marginal distribution and the extreme values of the field. Based on joint works with David Belius and Ron Peled.

*Title:* ** Scaling limit of the directed polymer on Z^{2+1} in the critical window**

*Abstract:* The directed polymer model on Z^{d+1} is the Gibbs transform of a directed random walk on Z^{d+1} in an i.i.d. random potential (disorder). It is known that the model undergoes a phase transition as the disorder strength varies, and disorder is relevant in d=1 and 2 in the sense that the presence of disorder, however weak, alters the qualitative behavior of the underlying random walk, with d=2 being the marginal case. For d=1, Alberts-Khanin-Quastel have shown that if the disorder strength tends to zero as a^{1/4} as the lattice spacing a tends to zero, then the partition functions converge to the solution of the Stochastic Heat Equation. We show that in the marginal dimension d=2, the partition functions admit non-trivial limits if the disorder strength scales as b/\sqrt{log 1/a}, with a transition at a critical point b_c. I will also discuss ongoing work in understanding the limit of the partition functions at b_c. Based on joint work with F. Caravenna and N. Zygouras.

*Title:* ** The Loewner Equation with Branching and the Continuum Random Tree**

*Abstract:* In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the “driving function” for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the scaling limit of these measures as the trees converge to the continuum random tree, with the goal of constructing an embedding of the CRT. We describe progress in this direction that has been obtained by analyzing the driving measures from an analytic standpoint, and we conclude by describing connections to the complex Burgers equation.

*Title:* ** Galton-Watson fixed points, tree automata, and interpretations**

*Abstract:* Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

*Title:* **On loops of Brownian motion**

*Abstract:* We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in (1, 5/3]. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

*Title:* **How round are the complementary components of planar Brownian motion?**

*Abstract:*Consider a Brownian motion W in the complex plane started from 0 and run for time 1. Let A(1), A(2),... denote the bounded connected components of C-W([0,1]). Let R(i) (resp.\ r(i)) denote the out-radius (resp.\ in-radius) of A(i) for i \in N. Our main result is that E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty for any \theta<1. We also prove that \sum_i r(i)^2|\log r(i)|=\infty almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

Friday, Mar 3: Wei Qian - ETH Zurich.

*Title:* **Decomposition of Brownian loop-soup clusters**

*Abstract:*We study the structure of Brownian loop-soup clusters in two dimensions.
The first part of the talk is based on joint-work with Wendelin Werner. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $l$ (which is known to be an SLE4-type loop), then the union of all excursions away from $l$ by all the Brownian loops in the loop-soup that touch $l$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $l$.
In the second part of the talk, we condition a Brownian loop-soup cluster (of any intensity) on a portion $p$ of its boundary and show that the union of loops that touch $p$ satisfies the restriction property. This result implies that a phase transition occurs at c = 14/15 for the connectedness of the loops that touch $p$.

*Title:* **Correlation inequalities for gradient fields and percolation**

*Abstract:* We consider a class of massless gradient Gibbs measures, in dimension greater or equal to three, with uniformly convex potential (and non-convex perturbations thereof). A well-known example in this class is the Gaussian free field, which has received considerable attention in recent years. We derive a so-called decoupling inequality for these fields, which yields detailed information about their geometry, and the percolative and non-percolative phases of their level sets. An important aspect is the development of a suitable sprinkling technique, interesting in its own right, which we will discuss in some detail. Roughly speaking, it allows to dominate the strong correlations present in the model, and crucially relies on a particular representation of these correlations in terms of a random walk in a dynamic random environment, due to Helffer and Sjöstrand.

* Title:* ** Random planar geometry**

* Title:* ** First-passage percolation in random planar lattices**

*Lecture 1:*** Taming infinities **

Monday, October 24, 2016, 4:30pm–5:30pm, Ryerson 251

*Abstract:* Some physical and mathematical theories have the unfortunate feature that if one takes them at
face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common
name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists
to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will
see how they have recently been used to make precise analytical statements about the solutions of some equations whose
meaning was not even clear until now.

*Lectures 2 and 3:* **The BPHZ theorem for stochastic PDEs **

Tuesday, October 25, 2016, 4:30pm–5:30pm, Eckhart 202

Wednesday, October 26, 2016, 4pm–5pm, Eckhart 202

*Abstract:* The Bogoliubov-Parasiuk-Hepp-Zimmermann theorem is a cornerstone of perturbative
quantum field theory: it provides a consistent way of "renormalising" the diverging integrals appearing
there to turn them into bona fide distributions. Although the original article by Bogoliubov and Parasiuk
goes back to the late 50s, it took about four decades for it to be fully understood. In the first lecture,
we will formulate the BPHZ theorem as a purely analytic question and show how its solution arises very
naturally from purely algebraic considerations. In the second lecture, we will show how a very similar
structure arises in the context of singular stochastic PDEs and we will present some very recent progress on
its understanding, both from the algebraic and the analytical point of view.

*First Lecture* : **Modeling and Estimating Massive Networks: Overview**

October 28, 4PM, Ryerson 251

*Second Lecture:* **Limits and Stochastic Models for Sparse Massive Networks**

October 31, 4PM, Eckhart 202

*Third Lecture:* ** Exchangeablity and Estimation of Sparse Massive Networks**

November 1, 4PM, Eckhart 206

* Title:* ** The Borell-Ehrhard game**

* Abstract: *A precise description of the convexity of Gaussian measures is provided by
a remarkable Brunn-Minkowski type inequality due to Ehrhard and Borell.
The delicate nature of this inequality has complicated efforts to develop
more general geometric inequalities in Gauss space that mirror the rich
family of results in the classical Brunn-Minkowski theory. In this talk, I
will aim to shed some new light on Ehrhard's inequality by showing that it
arises from a somewhat unexpected game-theoretic mechanism. This insight
makes it possible to identify new results, such as an improved form of
Barthe's reverse Brascamp-Lieb inequality in Gauss space. If time permits,
I will also outline how probabilistic ideas enabled us (in work with Yair
Shenfeld) to settle the equality cases in the Ehrhard-Borell inequalities.

* Title:* **Percolative properties of Brownian interlacements and its vacant set**

* Abstract: *In this talk, I will give a brief introduction to Brownian interlacements,
and investigate various percolative properties regarding this model. Roughly
speaking, Brownian interlacements can be described as a certain Poissonian cloud of
doubly-infinite continuous trajectories in the d-dimensional Euclidean space, d
greater or equal to 3, with the intensity measure governed by a level parameter. We
are interested in both the interlacement set, which is an enlargement (“the
sausages”) of the union of the trace in the aforementioned cloud of trajectories,
and the vacant set, which is the complement of the interlacement set. I will talk
about the following results: 1) The interlacement set is “well-connected”, i.e., any
two “sausages” in d-dimensional Brownian interlacements, can be connected via no
more than ceiling((d − 4)/2) intermediate sausages almost surely. 2) The vacant set
undergoes a non-trivial percolation phase transition when the level parameter
varies.

* Title:* ** Conformally invariant loop measures**

* Abstract: *We will discuss several aspects of a conjecture by Kontsevich
and Suhov regarding existence and uniqueness of a one parameter family of
conformally invariant measures on simple loops (conjecturally related to
the SLE family). The most natural case (zero central charge i.e. SLE
parameter kappa=8/3) was understood in a paper of Werner predating the
conjecture. In a work in progress, Dubédat and myself construct loop
measures in the whole conjectural range of existence (i.e. parameters kappa
for which SLE is a simple curve).

* Title:* ** Twin peaks**

* Abstract: * I will discuss some questions and results on random labelings of
graphs
conditioned on having a small number of peaks (local maxima).
The main open question is to estimate the distance between two peaks
on a large discrete torus, assuming that the random labeling
is conditioned on having exactly two peaks.
Joint work with Sara Billey, Soumik Pal, Lerna Pehlivan and Bruce Sagan.

* Title:* ** Regularity structure theory and its applications**

* Abstract: * Stochastic PDEs arise as important models in probability and mathematical physics.
They are typically nonlinear, driven by very singular random forces. Due to lack of
regularity it is typically very challenging to even interpret what one means by a
solution. In this talk I will explain the solution theories for some of these
equations, with a focus on the theory of regularity structures recently developed by
Martin Hairer. As applications of these theories, one can make sense of the
solutions to these stochastic PDEs, and once their solution theories are established
various convergence or approximation problems can be tackled.

* Title:* **First passage percolation on the exponential of two-dimensional
branching random walk: subsequential scaling limit at high temperature**

* Abstract: * Abstract: Let \{\eta_{N, v}: v\in V_N\} be a branching random walk in a
two-dimensional box V_N of side length N, that is, a 4-ary BRW with
Gaussian increments indexed by lattice points (with approximately
log-correlated covariances). We study the first passage percolation metric
where each vertex v is given a random weight of e^{\gamma \eta_{N, v}}. I
will show that for sufficiently small but fixed \gamma>0, for any sequence
of \{N_k\} there exists a subsequence along which the appropriately scaled
FPP metric converges in the Gromov-Hausdorff sense to a random metric on
the unit square in R^2. In addition, all possible (conjecturally unique)
scaling limits are non-trivial and are continuous with respect to the
Euclidean metric. Joint work with J. Ding.

* Title:* **Delocalization and Universality of band matrices**

* Abstract: * In this talk we introduce our new work on band matrices, whose
eigenvectors and eigenvalues are widely believed to have the same asymptotic
behaviors as those of Wigner matrices. We proved that this belief is true as long as
the bandwidth is wide enough.

* Title:* ** Circle packing and uniform spanning forests of planar graphs**

* Abstract: *The Koebe-Andreev-Thurston Circle Packing Theorem lets us draw planar graphs in a
canonical way, so that the geometry of the drawing reveals analytic properties of
the graph. Circle packing has proven particularly effective in the study of random
walks on planar graphs, where it allows us to estimate various quantities in terms
of their counterparts for Brownian motion in the plane.
In this talk, I will introduce the theory of circle packing and discuss work with
Asaf Nachmias in which we use circle packing to study uniform spanning forests of
planar graphs, a probability model closely related to random walk. We prove that the
free uniform spanning forest of any bounded degree, proper planar graph is connected
almost surely, answering positively a question of Benjamini, Lyons, Peres and
Schramm.
Our proof is quantitative, and also shows that uniform spanning forests exhibit some
of the same behaviour universally for all bounded degree transient triangulations,
provided that one measures distances and areas in the triangulation using the
hyperbolic geometry of its circle packing rather than with the usual graph metric
and counting measure.

* Title:* ** An almost sure KPZ relation for SLE and Brownian motion**

* Abstract: *I will discuss a KPZ-type formula which relates the Hausdorff
dimension of any set associated with SLE, CLE, or related processes; and
the Hausdorff dimension of a corresponding set associated with a correlated
two-dimensional Brownian motion. In many cases, the dimension of the
Brownian motion set is already known or easy to compute. This gives rise to
new proofs of the dimensions of several sets associated with SLE, including
the SLE curve; the double points and cut points of SLE; and the
intersection of two flow lines of a Gaussian free field. The formula is
based on the peanosphere construction of Duplantier, Miller, and Sheffield
(2014), which encodes a Liouville quantum gravity (LQG) surface decorated
with an independent space-filling SLE curve by means of a correlated
two-dimensional Brownian motion. I will give a moderately detailed overview
of this construction. Based on a joint work with Nina Holden and Jason
Miller http://arxiv.org/abs/1512.01223.

* Title:* ** The Parisi variational problem**

* Abstract: *The Parisi Variational Problem is a challenging non-local, strictly convex
variational problem over the space of probability measures whose analysis
is of great interest to the study of mean field spin glasses. In this
talk, I present a conceptually simple approach to the study of
this problem using techniques from PDEs, stochastic optimal control, and
convex optimization. We begin with a new characterization of the
minimizers of this problem whose origin lies in the first order optimality
conditions for this functional. As a demonstration of the power of this
approach, we study a prediction of de Almeida and Thouless
regarding the validity of the 1 atomic anzatz. We generalize their
conjecture to all mixed p-spin glasses and prove that their condition is
correct in the entire temperature-external field plane except for a
compact set whose phase is unknown at this level of generality. A key
element of this analysis is a new class of estimates regarding gaussian
integrals in the large noise limit called ``Dispersive Estimates of
Gaussians’’ . This is joint work with Ian Tobasco (NYU Courant).

* Title:* ** Arm Exponents for SLE**

* Abstract: *In the study of lattice models, arm exponents play an important role. In
this talk, we first discuss the arm exponents for critical percolation,
explain how they are derived and why they are important.
Second, we introduce the arm exponents for chordal SLE and explain the application
to the critical Ising and FK-Ising model.
Finally, we give a brief idea on deriving these exponents and some related open
questions.

* Title:* **Convergence of naturally parametrized
loop-erased random walk to the Schramm-Loewner
evolution parametrized by Minkowski content**

* Abstract: *The main goal of this talk is to explain the title.
I will define the terms
(type of convergence, naturally parametrized, loop-erased
random walk, Schramm-Loewner evolution, Minkowski
content) as well as the result. This is based on work
with Fredrik Wiklund.

* Title:* **Evolution of one-cells on a line**

* Abstract: *We consider systems with the following description.
At time zero, the real line is partitioned into intervals.
The original partition, which may be random, evolves according
to a deterministic rule whereby the interface between consecutive
pair of cells move so that the larger cell grows and the
smaller cell shrinks. When a cell shrinks to zero it disappears
and the two bounding points coalesce.
I will discuss one such system: a somewhat degenerate
one-dimensional version of a two (and higher) dimensional
mean-curvature flow model about which almost nothing rigorous
is known. In joint work with Emanuel Lazar, we prove that
the Poisson measure is invariant for this evolution, provided
that space is rescaled exponentially. We do this by introducing
the dual process (time-reversal). This process, unlike the forward
process, contains some randomness and may be exactly analyzed.
A number of questions remain open, such as uniqueness of trajectories,
convergence to Poisson from other initial conditions, and stability
under perturbation. Finally, I will discuss other one-dimensional
models with similar descriptions about which even less is known.

* Title:* ** Correlation distillation in probability spaces**

* Abstract: *Given a finite exchangeable collection of random variables in a probability
space, the correlation distillation problem asks for the partition of the
space into sets of a given measure as to maximize the probability that all
random variables lie in the same set. This problem is closely related to
isoperimetric problems and is motivated by applications in voting,
theoretical computer science and information theory. In the talk I will
survey some older and some recent results on correlation distillation. Many
open problems will be presented.

* Title:* **Dimensionality Reduction Via Sparse Matrices**

* Abstract: *This talk will discuss sparse Johnson-Lindenstrauss
transforms, i.e. sparse linear maps into much lower dimension which
preserve the Euclidean geometry of a set of vectors. Both upper and
lower bounds will be presented, as well as applications to certain
domains such as numerical linear algebra and compressed sensing.
Based on various joint works with Jean Bourgain, Sjoerd Dirksen,
Daniel M. Kane, and Huy Le Nguyen.

* Title:* **Speed of random walks on Cayley graphs of finitely generated groups**

* Abstract: *In this walk I will discuss a new construction of a family of groups. We
show that up to an absolute constant factor, any function $f$ satisfying $f(1)=1$,
$f(n)/\sqrt{n}$, $n/f(n)$ both non-decreasing can be realized as speed function of
simple random walk on some finitely generated group. In particular, it implies any
number in [1/2,1] can be realized as the speed exponent of simple random walk on
some group. The construction is very flexible and allows us to answer positively a
recent conjecture of Gideon Amir regarding joint behavior of speed and entropy. We
evaluate the Hilbert compression exponents of the groups under consideration. In
particular, we show that for any $\alpha\in[2/3,1]$, there exists a 3-step solvable
group with Hilbert compression exponent $\alpha$. It follows that there exists
uncountably many pairwise non quasi-isometric finitely generated 3-step solvable
groups.
Joint work with Jeremie Brieussel.

* Title:* **SPDE techniques for the random conductance model**

* Abstract: *I will survey some of the recent work applying techniques from
partial differential equations to the random conductance model on the lattice.
This will include some work of mine with Armstrong and some work of
Armstrong-Kuusi-Mourat and Gloria-Otto. There are now two approaches to
obtaining optimal rates in stochastic homogenization in divergence form.
The first obtains Green's function estimates by appealing to the
Efron-Stein concentration inequality. The second uses regularity theory to
localize the dependence of the solution on the coefficients. I will
discuss both of these methods.

* Title:* **Almost Sure Multifractal Spectrum of SLE**

* Abstract: *15 years ago B. Duplantier predicted the multifractal spectrum of Schramm
Loewner Evolution (SLE), which encodes the fine structure of the harmonic
measure of SLE curves. In this talk, I will report our recent rigorous
derivation of this prediction. As a byproduct, we also confirm a conjecture
of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE.
The proof uses various couplings of SLE and Gaussian free field, which are
developed in the theory of imaginary geometry and Liouville quantum
gravity. (Joint work with E. Gwynne and J. Miller.)

* Title:* **The Hardy-Littlewood-Sobolev inequality via martingale transforms**

* Abstract: *We outline a martingale proof of the classical Hardy-Littlewood-Sobolev
(HLS) inequality which naturally extends to the setting of Markovian semigroups that
have finite dimension in the sense of Varopoulos. The motivation for this approach
comes from efforts to employ probabilistic techniques to study (extend) the sharp
HLS inequality of E.H.Lieb.

* Title:* **On multilevel Dyson Brownian motions**

* Abstract: *I will discuss how Dyson Brownian motions describing the
evolution of eigenvalues of random matrices can be extended to multilevel
Dyson Brownian motions describing the evolution of eigenvalues of minors of
random matrices. The construction is based on intertwining relations
satisfied by the generators of Dyson Brownian motions of different
dimensions. Such results allow to connect general beta random matrix theory
to particle systems with local interactions, and to obtain novel results
even in the case of classical GOE, GUE and GSE random matrix models. Based
on joint work with Vadim Gorin.

* Title:* **Isomorphism theorems for space-time random walks**

* Abstract: * Loop measures have become important in the analysis of random
walks and connected research in mathematical physics. Such measures go
back to Symanzik in the late 1960s in the context of Euclidean field
theory. We discuss loop measures on graphs with countable infinite
different time horizons. These measures are connected to the cycle
representation of partition functions in quantum systems (Boson systems).
We derive corresponding Dynkin isomorphism theorems for space-time random
walks and we prove for some specific models the onset of the so-called
Bose-Einstein condensation.

* Title:* ** Discrete Fractal Dimensions and Large Scale Multifractals**

* Abstract: *Ordinary fractal dimensions such as Hausdorff dimension and packing
dimension are useful for analyzing the (microscopic) geometric structures
of various thin sets and measures. For studying (macroscopic or global)
fractal phenomena of discrete sets, Barlow and Taylor (1989, 1992)
introduced the notions of discrete Hausdorff and packing dimensions.
In this talk we present some recent results on macroscopic multifractal
properties of random sets associated with the Ornstein-Uhlenbeck
process and the mild solution of the parabolic Anderson model. (Joint work
with Davar Khoshnevisan and Kunwoo Kim.)

* Title:* ** On multilevel Dyson Brownian motions.**

* Abstract: *I will discuss how Dyson Brownian motions describing the
evolution of eigenvalues of random matrices can be extended to multilevel
Dyson Brownian motions describing the evolution of eigenvalues of minors of
random matrices. The construction is based on intertwining relations
satisfied by the generators of Dyson Brownian motions of different
dimensions. Such results allow to connect general beta random matrix theory
to particle systems with local interactions, and to obtain novel results
even in the case of classical GOE, GUE and GSE random matrix models. Based
on joint work with Vadim Gorin.

* Title:* ** A simple renormalization flow setup for FK-percolation models**

* Abstract: *We will present a simple setup in which one can make sense of a renormalization flow
for FK-percolation models in terms of a simple Markov process on a state-sace of
discrete weighted graphs. We will describe how to formulate the universality
conjectures in this framework (in terms of stationary measures for this Markov
process), and how to prove this statement in the very special case of the
two-dimensional uniform spanning tree (building on existing results on this model).
This is based in part on joint work with Stéphane Benoist and Laure Dumaz.

* Title:* ** Dynamics on random regular graphs: Dyson Brownian motion and the Poisson free field**

* Abstract: *: A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider d many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree 2d. Consider the problem of eigenvalue fluctuations of the adjacency matrix of such a graph. We consider the following dynamics. The ‘dimension’ of each permutation grows by coupled Chinese Restaurant Processes, while in ‘time’ each permutation evolves according to the random transposition Markov chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. We give a Poisson random surface description in dimension and time of the limiting cycle counts for every d. As d grows to infinity, these Poisson random surfaces converge to the Gaussian Free Field preserved in time by the Dyson Brownian motion. Part of this talk is based on a joint work with Tobias Johnson and the rest is based on a joint work with Shirshendu Ganguly. (Cambridge).

* Title:* ** Maxima of log-correlated Gaussian fields and of the Riemann Zeta
function**

* Abstract: * A recent conjecture of Fyodorov, Hiary & Keating states that the
maxima of the Riemann Zeta function on a bounded interval of the critical
line behave similarly to the maxima of a specific class of Gaussian fields,
the so-called log-correlated Gaussian fields. These include important
examples such as branching Brownian motion and the 2D Gaussian free field.
In this talk, we will highlight the connections between the number theory
problem and the probabilistic models. We will outline the proof of the
conjecture in the case of a randomized model of the Zeta function. We will
discuss possible approaches to the problem for the function itself. This is
joint work with D. Belius (NYU) and A. Harper (Cambridge).

* Title:* ** Random matrices have simple spectrum**

* Abstract: *A symmetric matrix has simple spectrum if all eigenvalues are different.
Babai conjectured that random graphs have simple spectrum with probability tending
to 1.
Confirming this conjecture, we prove the simple spectrum property for a large class
of random matrices.
If time allows, we will discuss the harder problem of bounding the spacings between
consecutive eigenvalues, with applications in mathematical physics, computer
science, and numerical linear algebra.
Several open questions will also be presented.
Joint work with H. Nguyen (OSU) and T. Tao (UCLA).

* Title:* ** Exactly solvable mean-field monomer-dimers models**

* Abstract: *A The seminar will introduce some mean-field models
used to describe monomer-dimers systems. In
particular the solution for the diluted case and the
random impurity case will be shown and the absence
of phase transition proved.

* Title:* ** A local central limit theorem for random representations of SU(3)**

* Abstract: *The number p(n) of integer partitions of n is given approximately for large n by a
famous asymptotic formula proved by Hardy and Ramanujan in 1918. This can be
interpreted as a statement about the number of inequivalent representations of
dimension n of the group SU(2). In this talk I will discuss my recent proof of an
analogous result for the asymptotic number of n-dimensional representations of the
group SU(3). A key step is to prove a local central limit theorem in a probabilistic
model for random representations, which requires some ideas from the theory of
modular forms. I will explain these ideas, as well as connections to a mysterious
“Witten zeta function" associated with SU(3), and additional applications to
understanding the limit shape of random n-dimensional representations of SU(3). No
knowledge of representation theory will be assumed or needed.

* Title:* **Effect of initial conditions on mixing for the Ising Model**

* Abstract: *Recently, the ``information percolation'' framework was introduced as a way
to obtain sharp estimates on mixing for the high temperature Ising model,
and in particular, to establish cutoff in three dimensions up to
criticality from a worst starting state. I will describe how this method
can be used to understand the effect of different initial states on the
mixing time, both random (''warm start'') and deterministic.
Joint work with Allan Sly.

* Title:* **Proportional Switching in FIFO Networks**

* Abstract: * A central problem in queueing theory is the development of
policies that efficiently allocate available resources. Many standard
policies have a fixed capacity at individual sites, rather than the ability
to allocate resources across sites. We discuss here the proportional
switching policy, where the amount of service at different sites is
dependent and the corresponding service vector is required to lie in a
convex region. We also assume that packets are served in the FIFO
(first-in, first-out) order. Past work on the stability of proportional
switching networks has focused on networks with elementary routing
structure (such as immediate departure after service at a site). Here, we
consider the stability problem for general routing structures. The talk
is based on joint work with B. D'Auria and N. Walton.

3:35-4:35 Friday, June 5th: Paul Jung -University of Alabama Birmingham.

* Title:* **Levy Khintchine random matrices and the Poisson weighted infinite skeleton
tree**

* Abstract: * We study a class of Hermitian random matrices which includes Wigner
matrices, heavy-tailed random matrices, and sparse random matrices such as
adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real
entries which are i.i.d. up to symmetry. The distribution of entries
depends on N, and we require sums of rows to converge in distribution; it
is then well-known that the limit must be infinitely divisible.
We show that a limiting empirical spectral distribution (LSD) exists, and
via local weak convergence of associated graphs, the LSD corresponds to the
spectral measure associated to the root of a graph which is formed by
connecting infinitely many Poisson weighted infinite trees using a backbone
structure of special edges. One example covered are matrices with i.i.d.
entries having infinite second moments, but normalized to be in the
Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

* Title:* **Recent results on the multispecies coalescent**

* Abstract: *The multispecies coalescent is a variant of Kingman’s coalescent in which several populations are stitched together on a base tree. Increasingly, it plays an important role in phylogenetics where it can be used to model the joint evolution of a large number of genes across multiple species. Motivated by information-theoretic questions, I will present a recent probabilistic analysis of the multispecies coalescent which establishes fundamental limits on the inference of this model from molecular sequence data. No biology background is required. This is joint work with Gautam Dasarathy, Elchanan Mossel, Rob Nowak, and Mike Steel.

* Title:* **Nash Equilibria for a Quadratic Voting Game**

* Abstract: *Voters making a binary decision purchase votes from a centralized clearing house,
paying the square of the number of votes purchased. The net payoff to an agent with
utility u who purchases v votes is \Psi(S)u−v^2, where \Psi is an odd, monotone
function taking values between -1 and +1 and S is the sum of all votes purchased by
the n voters participating in the election. The utilities of the voters are assumed
to arise by random sampling from a probability distribution F with compact support;
each voter knows her own utility, but not those of the other voters, although she
does know the sampling distribution F. Nash equilibria for this game are described.

* Title:* **Brownian motion on spaces with varying dimension**

* Abstract: * The model can be picturized as the random movement of an insect
on the ground with a pole standing on it. That is, part of the state space
has dimension 2, and the other part of the state space has dimension 1. We
define such a process as a ``darning process'' in terms of Dirichlet form,
because 2-dimensional Brownian motion does not hit any singleton. We show
that the behavior of this process switches between 1-dimensional and
2-dimensional, which depends on both the time and the positions of the
points. An open ongoing project will also be introduced: Can we approximate
such a process by random walks? The main results of this talk are based on
my joint work with Zhen-Qing Chen.

* Title:* **Constant Factor Approximation for Balanced Cut in the PIE Model**

* Abstract: * We propose and study a new semi-random semi-adversarial model
for Balanced Cut, a planted model with permutation-invariant random
edges (PIE). Our model is much more general than planted and stochastic
models considered previously. Consider a set of vertices V
partitioned into two clusters L and R of equal size. Let G be an
arbitrary graph on V with no edges between L and R. Let E_random be a
set of edges sampled from an arbitrary permutation-invariant
distribution (a distribution that is invariant under
permutation of vertices in L and in R). Then we say that G +
E_random is a graph with permutation-invariant random edges.
We present an approximation algorithm for the Balanced Cut problem
that finds a balanced cut of cost O(|E_random|) + n polylog(n) in this
model. In the regime when there are at least \Omega(n polylog(n)) random
edges, this is a constant factor approximation with respect to the
cost of the planted cut.
Joint work with: Konstantin Makarychev and Aravindan Vijayaraghavan.

* Title:* **Regularization under diffusion and Talagrand's convolution
conjecture**

* Abstract: *It is a well-known phenomenon that functions on Gaussian space become
smoother under the Ornstein-Uhlenbeck semigroup. For instance, Nelson's
hypercontractive inequality shows that if p > 1, then L^p functions are
sent to L^q functions for some q > p. In 1989, Talagrand conjectured* that
quantitative smoothing is achieved even for functions which are only L^1,
in the sense that under the semigroup, such functions have tails that are
strictly better than those predicted by Markov's inequality and
preservation of mass. Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff
(2010) proved that this holds in fixed dimensions. We resolve Talagrand's
conjecture conjecture positively (with no dimension dependence).
The key insight is to study a subset of Gaussian space at various
granularities by approaching it as "efficiently" as possible. To this end,
we employ an Ito process that arose in the context of optimal control
theory. Efficiency is measured by the average "work" required to couple
the approach process to a Brownian motion.
*Talagrand's full conjecture is for functions on the discrete cube. Here
we address the Gaussian limiting case.
This is joint work with Ronen Eldan.

* Title:* ** On the chemical distance in critical percolation**

* Abstract: *In two-dimensional critical percolation, the works of Aizenman-Burchard
and Kesten-Zhang imply that macroscopic distances inside percolation
clusters are bounded below by a power of the Euclidean distance greater
than 1+ε, for some positive ε. No more precise lower bound has been
given so far. Conditional on the existence of an open crossing of a box
of side length n, there is a distinguished open path which can be
characterized in terms of arm exponents: the lowest open path crossing
the box. This clearly gives an upper bound for the shortest path. The
lowest crossing was shown by Zhang and Morrow to have volume n^4/3+o(1)
on the triangular lattice.
Following a question of Kesten and Zhang, we compare the length of
shortest circuit in an annulus to that of the innermost circuit (defined
analogously to the lowest crossing). I will explain how to show that the
ratio of the expected length of the shortest circuit to the expected
length of the innermost crossing tends to zero as the size of the
annulus grows.
Joint work with Jack Hanson and Michael Damron.

* Title:* ** High temperature limits for $(1+1)$-d directed polymer with heavy-tailed
disorder.**

* Abstract: *The directed polymer model at intermediate disorder regime was introduced
by Alberts-Khanin-Quastel (2012). It was proved that at inverse temperature $\beta
n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately,
converges in distribution and the limit is given in terms of the solution of the
stochastic heat equation. This result was obtained under the assumption that the
disorder variables posses exponential moments, but its universality was also
conjectured under the assumption of six moments. We show that this conjecture is
valid and we further extend it by exhibiting classes of different universal limiting
behaviors in the case of less than six moments. We also explain the behavior of the
scaling exponent for the log-partition function under different moment assumptions
and values of $\gamma$. Based on joint work with Nikos Zygouras.

* Title:* ** Stochastic flows for Levy processes with Holder drifts**

* Abstract: *In this talk I will present some new results on the following
SDE in $R^d$:
$$
dX_t=b(t, X_t)dt+dZ_t, \quad X_0=x,
$$
where $Z$ is a Levy process. We show that for a large class
of Levy processes $Z$ and Holder continuous drfit $b$, the
SDE above has a unique strong solution for every starting point $x\in R^d$.
Moreover, these strong solutions form a $C^1$-stochastic flow. In
particular, we show that, when $Z$ is a symmetric $\alpha$-stable process
with $\alpha\in (0, 1]$ and $b$ is $\beta$-Holder continuous with
$\beta\in (1-\alpha/2, 1)$, the SDE above has a unique strong solution.

* Title:* ** Universality in spin glasses**

* Abstract: *This talk is concerned about some universal properties of the
Parisi solution in spin glass models. We will show universality of chaos
phenomena and ultrametricity in the mixed p-spin model under mild moment
assumptions on the environment. We will explain that the results also
extend to quenched self-averaging of some physical observables in the mixed
p-spin model as well as in different spin glass models including the
Edwards-Anderson model and the random field Ising model.

* Title:* **Displacement convexity of entropy and curvature in discrete settings**

* Abstract: *Inspired by exciting developments in optimal transport and Riemannian
geometry (due to the work of Lott-Villani and Sturm), several
independent groups have formulated a (discrete) notion of curvature in
graphs and finite Markov chains. I will describe some of these
approaches briefly, and mention some related open problems of
potential independent interest.

* Title:* **Finitely Dependent Coloring on Z and other Graphs**

* Abstract: *
In 2008, Oded Schramm asked the following question:
For what values of $k$ and $q$ does there exist a stationary, proper,
$k-$dependent $q-$coloring of the integers?
Schramm had a substantial amount of evidence, which I will describe, that convinced him that
such a coloring does not exist for any values of $k$ and $q$. In fact, it turns out that such an object
does exist for many values of $k$ and $q$. I will tell you exactly which ones
work, and will describe colorings with these properties. No
knowledge of advanced probability is needed to follow the lecture.
There are several connections with
combinatorics, but again, no specialized knowledge is needed.
This is joint work with A. Holroyd.

* Title:* **Stationary Eden Model on amenable groups**

* Abstract:
*We consider stationary versions of the Eden model, on a product of a Cayley graph G of an amenable group and positive integers.
The process results in a collection of disjoint trees rooted at G, each of which consists of geodesic paths in a corresponding first passage percolation model on the product graph. Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Haggstrom on the Euclidean lattice. This is a joint work with Eviatar Procaccia.

* Title:* **Random tilings and Hurwitz numbers**

* Abstract:
*This talk is about random tilings of a special class of planar domains,
which I like to call "sawtooth domains." Sawtooth domains have the special
feature that their tilings are in bijective correspondence with Gelfand-Tsetlin
patterns, aka semistandard Young tableaux. Consequently, many observables can be
expressed in terms of special functions of representation-theoretic origin. In
particular, the distribution of tiles of one type along a horizontal slice through
a uniformly random tiling is encoded by the Harish-Chandra/Itzykson-Zuber
integral, a familiar object from random matrix theory which also happens to be a
generating function for a desymmetrized version of the Hurwitz numbers from
enumerative algebraic geometry. I will explain how this fact allows one to prove
that tiles along a slice fluctuate like the eigenvalues of a Gaussian random
matrix.

* Title:* ** Lengths of Monotone Subsequences in a Mallows Permutation**

* Abstract:
*The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution.
We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q.
This is joint work with Ron Peled.

* Title:* **Rate of convergence of the mean for sub-additive ergodic sequences**

* Abstract:
*For a subadditive ergodic sequence {X_{m,n}}, Kingman's theorem gives convergence
for the terms X_{0,n}/n to some non-random number g. In this talk, I will discuss
the convergence rate of the mean EX_{0,n}/n to g. This rate turns out to be
related to the size of the random fluctuations of X_{0,n}; that is, the variance
of X_{0,n}, and the main theorems I will present give a lower bound on the
convergence rate in terms of a variance exponent. The main assumptions are that
the sequence is not diffusive (the variance does not grow linearly) and that it
has a weak dependence structure. Various examples, including first and last
passage percolation, bin packing, and longest common subsequence fall into this
class. This is joint work with Michael Damron and Jack Hanson.

* Title:* **Hierarchical approximations to the Gaussian free field and fast simulation
of Schramm-Loewner evolutions **

* Abstract:
*The Schramm--Loewner evolutions (SLE) are a family of stochastic processes which
describe the scaling limits of curves which occur in two-dimensional critical
statistical physics models. SLEs have had found great success in this task, greatly
enhancing our understanding of the geometry of these curves. Despite this, it is
rather difficult to produce large, high-fidelity simulations of the process due to
the significant correlation between segments of the simulated curve. The standard
simulation method works by discretizing the construction of SLE through the Loewner
ODE which provides a quadratic time algorithm in the length of the curve.
Recent work of Sheffield and Miller has provided an alternate description of SLE,
where the curve generated is taken to be a flow line of the vector field obtained by
exponentiating a Gaussian free field. In this talk, I will describe a new
hierarchical method of approximately sampling a Gaussian free field, and show how
this allows us to more efficiently simulate an SLE curve. Additionally, we will
briefly discuss questions of the computational complexity of simulating SLE which
arise naturally from this work.

* Title:* **Limited choice and randomness in the evolution of networks**

* Abstract: * The last few years have seen an explosion in network models describing the
evolution of real world networks. In the context of math probability, one aspect
which has seen an intense focus is the interplay between randomness and limited
choice in the evolution of networks, ranging from the description of the emergence
of the giant component, the new phenomenon of ``explosive percolation'' and power of
two choices. I will describe ongoing work in understanding such dynamic network
models, their connections to classical constructs such as the standard
multiplicative coalescent and applications of these simple models in fitting retweet
networks in Twitter.

* Title:* **An example of hypoellipticity in infinite dimensions**

* Abstract: * A collection of vector fields on a manifold satisfies H\"{o}rmander's condition if
any two points are connected by a path whose tangent vectors only lie in the given
directions.
It is well-known that a diffusion which is allowed to travel only in
these directions is smooth, in the sense that its transition probability measure is
absolutely continuous with respect to the volume measure and has a strictly positive
smooth density.
Smoothness results of this kind in infinite dimensions are typically not known, the
first obstruction being the lack of an infinite-dimensional volume measure. We will
discuss recent results on a particular class of infinite-dimensional spaces, where
we have shown that vector fields satisfying H\"{o}rmander's condition generate a
diffusion which has a strictly positive smooth density with respect to an
appropriate reference measure.

(**Mathematics Colloquium, 2:00 -3:00 pm @ Eckhart 206 **)

* Title:* ** Free probability and random matrices; from isomorphisms to universality**

* Abstract: *Free probability is a probability theory for non-commutative variables introduced by Voiculescu about thirty years ago. It is equipped with a notion of freeness very similar to independence. It is a natural framework to study the limit of random matrices with size going to infinity. In this talk, we will discuss these connections and how they can be used to adapt ideas from classical probability theory to operator algebra and random matrices. We will in particular focus on how to adapt classical ideas on transport maps following Monge and Ampere to construct isomorphisms between algebras and prove universality in matrix models. This talk is based on joint works with F. Bekerman, Y. Dabrowski, A. Figalli and D. Shlyakhtenko.

* Title:* **Strict Convexity of the Parisi Functional**

* Abstract: *Spin glasses are magnetic systems exhibiting both quenched disorder and frustration,
and have often been cited as examples of "complex systems." As mathematical objects,
they provide several fascinating structures and conjectures. This talk will cover
recent progress that shed more light in the mysterious and beautiful solution
proposed 30 years ago by G. Parisi. We will focus on properties of the free energy
of the famous Sherrington-Kirkpatrick model and we will explain a recent proof of
the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo
Chen.

(2:30-3:30) Elton P. Hsu - Northwestern University.

* Title:* **Brownian Motion and Gradient Estimates of Positive Harmonic Functions**

* Abstract: *Many gradient estimates in differential geometry can be naturally treated
by stochastic methods involving Brownian motion on a Riemannian manifold. In this
talk, we discuss Hamilton\'92s gradient estimate of bounding the gradient of the
logarithm of a positive harmonic function in terms of its supremum from this point
of view. We will see how naturally this gradient estimate follows from Ito\'92s
formula and extend it to manifolds with boundary by considering reflecting Brownian
motion. Furthermore, we will show that in fact Hamilton\'92s gradient estimate can
be embedded as the terminal case of a family of gradient estimates which can be
treated just as easily by the same stochastic method.

(4:00-5:00) Marek Biskup -UCLA.

* Title:* **Isoperimetry for two dimensional supercritical percolation**

* Abstract: *Isoperimetric problems have been around since ancient history. They play
an important role in many parts of mathematics as well as sciences in general.
Isoperimetric inequalities and the shape of isoperimetric sets are generally well
understood in Euclidean or other "nice" settings but are still subject of research
in random domains, graphs, manifolds, etc. In my talk I will address the
isoperimetric problem for one example of a random setting: the unique infinite
connected component of supercritical bond percolation on the square lattice. In
particular, I will sketch a proof of the fact that, as the volume of a (properly
defined) isoperimetric set tends to infinity, its asymptotic shape can be
characterized by an isoperimetric problem in the plane with respect to a particular
(continuum) norm. As an application I will conclude that that the anchored
isoperimetric profile with respect to a given point as well as the Cheeger constant
of the giant component in finite boxes scale to deterministic quantities. This
settles a conjecture of Itai Benjamini for the plane. Based on joint work with O.
Louidor, E. Procaccia and R. Rosenthal.

* Title:* **Continuous spectra for sparse random graphs**

* Abstract: *The limiting spectral distributions of many sparse random graph models
are known to contain atoms. But a more interesting question is when they
also have some continuous part. In this talk, I will give affirmative
answer to this question for several widely studied models of random graphs
including Erdos-Renyi random graph G(n,c/n) with c > 1, random graphs with
certain degree distributions and supercritical bond percolation on Z^2. I
will also present several open problems.
This is joint work with Charles Bordenave and Balint Virag.

* Title:* ** Snowflakes, slot machines, Chinese dragons, and QLE **

*Abstract:* What is the right way to think of a "random surface" or a "random planar
graph"? How can one explain the dendritic patterns that appear in snowflakes,
choral reefs, lightning bolts, and other physical systems, as well in as toy
mathematical models inspired by these systems? How are these questions related to
random walks and random fractal curves (in particular the famous SLE curves)?
To begin to address these questions, I will introduce and explain the "quantum
Loewner evolution", which is a family of growth processes closely related to SLE. I
will explain. through pictures and animations and some discrete arguments, how QLE
is defined and what role it might play in addressing the questions raised above.
In a continuation of the talk on Friday afternoon (at the probability seminar), I
will present a more analytic, continuum construction of QLE and discuss its
relationship to the so-called Brownian map.
Joint work with Jason Miller.

* Title:* **Two-sided radial SLE and length-biased chordal SLE**

* Abstract: * Models in statistical physics often give measures on self-avoiding
paths. We can restrict such a measure to the paths that pass through a
marked point, obtaining a "pinned measure". The aggregate of the
pinned measures over all possible marked points is just the original
measure biased by the path's length. Does the analogous result hold
for SLE curves, which appear in the scaling limits of many such models
at criticality? We show that it does: the aggregate of two-sided
radial SLE is length-biased chordal SLE, where the path's length is
measured in the natural parametrization.

* Title:* **Planar growth models and conformal mapping**

* Abstract: * Random, fractal-like growth can be seen in several places in nature.
Several mathematical models based in one way or another on harmonic measure exist, but despite significant efforts little is known about these models.
I will survey some of the models and problems, focusing in particular on constructions based on conformal maps.
Towards the end I will discuss some recent joint work with Sola and Turner on one of these models.

Edward Waymire - Oregon state university. (2:30-3:30)

* Title:* **Tree Polymers Under Strong Disorder**

* Abstract: * Tree polymers are simplifications of
1+1 dimensional lattice polymers made up
of polygonal paths of a (nonrecombining)
binary tree having random path probabilities.
The path probabilities are (normalized) products of i.i.d.
positive weights. As such, they reside in the more general
framework of multiplicative cascades and branching random
walk. The probability laws of these paths are of interest under
weak and strong types of disorder. Some recent results, speculation
and conjectures will be presented for this class of models under both
weak and strong disorder conditions. This is based on various joint
papers with Partha Dey, Torrey Johnson, or Stan Williams.

Wei Wu - Brown University. (3:35-4:35)

* Title:* **Random fields from uniform spanning trees**

* Abstract: * The uniform spanning tree (UST) is a fundamental combinatorial
object. In two dimensions, using conformal invariance and planar duality,
it is shown that the scaling limits of UST is given by one of the SLE path.
We discuss the random field approach, and study the scaling limit of
certain random fields coupled with USTs. This approach works on general
graphs, and may help to understand the scaling limits of UST in higher
dimensions. This talk is based on several joint works with Adrien Kassel,
Richard Kenyon and Xin Sun.

* Title:* **Intersections of SLE paths**

* Abstract: * SLE curves are introduced by Oded Schramm as the candidate of the
scaling limit of discrete models. In this talk, we first describe basic
properties of SLE curves and their relation with discrete models. Then we
summarize the Hausdorff dimension results related to SLE curves, in
particular the new results about the dimension of cut points and double
points. Third we introduce Imaginary Geometry, and from there give the idea
of the proof of the dimension results.

* Title:* **On Gaussian inequalities for product of functions**

* Abstract: * Gaussian inequalities have played important roles in various scientific
areas. In this talk, we will present simple algebraic criteria that yield
sharp Holder types of inequalities for the product of functions of Gaussian
random vectors with arbitrary covariance structure. As an application, we
will explain how our results yield several famous inequalities in
functional geometry, such as, the Brascamp-Lieb inequality, the sharp Young
inequality, etc. This part of the talk is based on the recent joint work
with N. Dafnis and G. Paouris. Along this direction, we will discuss a
conjecture on the convexity of the Parisi functional arising from the study
of the Sherrington-Kirkpatrick model in spin glass.

* Title:* **Conformal invariance of the Green's function for
loop-erased random walk**

* Abstract: * The planar loop-erased random walk (LERW) is obtained from
the usual random walk by erasing loops. The LERW is
related to a number of other models such as the
uniform spanning tree. We consider a fixed simply connected
domain in C containing the origin, and two distinct boundary points
a and b. For a fixed lattice spacing, we consider the probability
that a LERW goes from a to b goes through an edge containing
the origin. We show that the normalized limit of this probability
goes to a conformally covariant quantity, the Green's function
for the Schramm-Loewner evolution.
This is joint work with Christian Benes and Fredrik Viklund
.

* Title:* **Random walks on planar graphs via circle packings**

* Abstract: *I will describe two results concerning random walks on planar graphs and
the connections with Koebe's circle packing theorem (which I will not
assume any knowledge of):
1. A bounded degree planar triangulation is recurrent if an only if the set
of accumulation points of its circle packing is a polar set (that is, has
zero logarithmic capacity). This extends a result of He and Schramm who
proved recurrence (transience) when the set of accumulation points is empty
(a closed Jordan curve). Joint work with Ori Gurel-Gurevich and Juan Souto.
2. The Poisson boundary (the space of bounded harmonic functions) of a
transient bounded degree triangulation of the plane is characterized by the
topological boundary obtained by circle packing the graph in the unit disk.
In other words, any bounded harmonic function on the graph is the harmonic
extension of some measurable function on the boundary of the unit disc.
Joint work with Omer Angel, Martin Barlow and Ori Gurel-Gurevich.

* Title:* **A Two-Sided Estimate for the Gaussian Noise Stability Deficit **

* Abstract: *The Gaussian Noise Stability of a set A in Euclidean space is the probability that for
a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borell's celebrated inequality states that
a half-space maximizes the noise stability among all possible sets having the same Gaussian measure. We present a novel short proof of this
inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality:
We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance
between the corresponding centroids. As a consequence, we prove a conjecture of Mossel and Neeman, who used the total-variation distance as a metric.

* Title:* **Asymptotic behavior of log-concave probability measures**

* Abstract: * A probability measure $\mu$ in ${\mathbb R}^n$ is called log-concave if
$\mu\big(\lambda A + (1-\lambda) B\big) \geq \mu(A)^\lambda\,\mu(B)^{1-\lambda}$,
for every $\lambda\in[0,1]$ and every $A,B$ Borel subsets of ${\mathbb R}^n$.
Two basic examples are the uniform measure restricted to a convex body
in ${\mathbb R}^n$ with volume $1$ (Brunn-Minkowski inequality) and the normal
Gaussian measure in ${\mathbb R}^n$.
We are studying the asymptotic behavior of some random geometric quantities
such as the volume and the radius of a random polytope generated by sampling
with respect to a log-concave probability measure.
We will show that asymptotically ( as the dimension $n$ goes to infinity),
they behave like if we had sampled with respect to the Gaussian measure.

* Title:* ** Performance of the Metropolis algorithm on a disordered tree: the Einstein relation.**

* Abstract: * Consider a d-ary rooted tree (d>2) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S_n* of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algorithm on the tree and show that under these assumptions there always exists a temperature of the algorithm so that it achieves a linear (positive) growth rate in linear time. This confirms a conjecture of Aldous (Algorithmica, 22(4):388-412, 1998). The proof is obtained by establishing an Einstein relation for the Metropolis algorithm on the tree. Joint work with Pascal Maillard.

* Title:* ** Persistence Probabilities. **

* Abstract: * Persistence probabilities concern how likely it is that a stochastic
process has a long excursion above fixed level and of what are the
relevant scenarios for this behavior. Power law decay is expected
in many cases of physical significance and the issue is to determine
its power exponent parameter. I will survey recent progress in this
direction (jointly with Sumit Mukherjee), dealing with stationary
Gaussian processes that arise from random algebraic polynomials
of independent coefficients and from the solution to heat
equation initiated by white noise.
If time permits, I will also discuss the relation to joint works
with Jian Ding and Fuchang Gao, about persistence for iterated
partial sums and other auto-regressive sequences, and to the
work of Sakagawa on persistence probabilities for the height
of certain dynamical random interface models.

* Title:* ** Statistics on Hilbert's Sixteenth Problem **

* Abstract: * The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.

* Title:* ** A Dynamic Graph Model of Barter Exchanges **

* Abstract: * Motivated by barter exchanges, we study average waiting time in a dynamic random graph model. A node arrives at each time step. A directed edge is formed independently with probability p with each node currently in the system. If a cycle is formed, of length no more than 3, then that cycle of nodes is removed immediately. We show that the average waiting time for
> a node scales as 1/p^{3/2} for small p, for this policy. Moreover, we prove that we cannot achieve better delay scaling by batching.
Our results through new light on the operation of kidney exchange programs. The insight offered by our analysis is that the beneﬁt of waiting for additional incompatible patient-donor pairs to arrive (batching) into kidney exchange clearinghouses is not substantial and is outweighed by the cost of waiting.
Joint work with Ross Anderson, Itai Ashlagi and David Gamarnik.

* Title:* ** Epsilon-biased sets, the Legendre symbol, and getting by with a few random bits **

* Abstract: * Subsets of F_2^n that are p-biased, meaning that the parity of any set of bits is even or odd with probability close to 1/2, are useful tools in derandomization. They also correspond to optimal error-correcting codes,i.e. meeting the Gilbert-Varshamov bound, with distance close to n/2.
A simple randomized construction shows that such sets exist of size O(n/p^2); recently, Ben-Aroya and Ta-Shma gave a deterministic construction of size O((n/p^2)^(5/4)). I will review deterministic constructions of Alon, Goldreich, Haastad, and Peralta of sets of size
O(n/p^3) and O(n^2/p^2), and discuss the delightful pseudorandom properties of the Legendre symbol along the way.
Then, rather than derandomizing these sets completely in exchange for making them larger, we will try moving in a different direction on the size-randomness plane, constructing sets of optimal size O(n/p^2) with as few random bits as possible. The naive randomized construction requires O(n^2/p^2) random bits. I will show that this can be reduced to O(n log(n/p)) random bits. Like Alon et al., our construction uses the Legendre symbol and Weil sums, but in a different way to control high moments of the bias. I'll end by saying a few words about Ramsey graphs and random polynomials.
This is joint work with Alex Russell.

* Title:* ** Quantum spin systems and graphical representations **

* Abstract: * Quantum spin systems are mathematical models for magnetism. But the quantum nature is a difficulty. For some models there are graphical representations, which relate to interacting particle processes (with some changes). I will discuss one application done jointly with Nick Crawford and Stephen Ng, called emptiness formation probability where this approach works.

* Title:* **Delocalization of eigenvectors of random matrices **

* Abstract: * Eigenvectors of random matrices are much less studied than eigenvalues, despite their importance. The simplest question is whether the eigenvectors are delocalized, i.e. all of their coordinates are as small as can be, of order n^{-1/2}. Even this simple looking problem has been open until very recently. Currently there are two approaches to delocalization - spectral (via local eigenvalue statistics) and geometric (via high dimensional probability). This talk will explain these approaches and popularize related open problems. Based on joint work with Mark Rudelson (Michigan).

* Title:* ** Multiple Phase Transitions for long range first-passage percolation on lattices **

* Abstract: * Given a graph G with non-negative edge weights, the passage time of a path is
the sum of weights of the edges in the path, and the first-passage time to reach u from v is the
minimum passage time of a path joining them. We consider a long range first-passage model
on Z^d in which, the weight w(x,y) of the edge joining x and y has exponential distribution
with mean |x-y|^a for some fixed a > 0, and the edge weights are independent. We analyze
the growth of the set of vertices reachable from the origin within time t, and show that there
are four different growth regimes depending on the value of a.
Joint work with Partha Dey.

* Title:* ** Statistics on Hilbert's Sixteenth Problem **

* Abstract: * The first part of Hilbert's sixteenth problem concerns real algebraic geometry: We are asked to study the number and possible arrangements of the connected components of a real algebraic curve (or hypersurface). I will describe a probabilistic approach to studying the topology, volume, and arrangement of the zero set (in real projective space) of a random homogeneous polynomial. The outcome depends on the definition of "random". A popular Gaussian ensemble uses monomials as a basis, but we will favor eigenfunctions on the sphere (spherical harmonics) as a basis. As we will see, this "random wave" model produces a high expected number of components (a fraction of the Harnack bound that was an inspiration for Hilbert's sixteenth problem). This is joint work with Antonio Lerario.

*Title:* **On the monotonicity of the speed of biaised random walk on a Galton-Watson tree without leaves.**

*Abstract:* We will present different results related to the speed of biased random walks in random environments. Our focus will be on a recent paper by Ben Arous, Fribergh and
Sidoravicius proving that the speed of the biased random walk on a Galton-Watson tree
without leaves is increasing for high biases. This partially solves a question asked by Lyons,
Pemantle and Peres.

*Title:* **Search Games, The Cauchy process and Optimal Kakeya Sets **

*Abstract: * A planar set that contains a unit segment in every direction is
called a Kakeya set. These sets have been studied intensively in
geometric measure theory and harmonic analysis since the work of
Besicovich (1928); we find a new connection to game theory and
probability via a search game first analyzed by Adler et al (2003). A
hunter and a rabbit move on the n-vertex cycle without seeing each other.
At each step, the hunter moves to a neighboring vertex or stays in place,
while the rabbit is free to jump to any node. Thus they are engaged in a
zero sum game, where the payoff is the capture time. The known optimal
randomized strategies for hunter and rabbit achieve expected capture time
of order n log n. We show that every rabbit strategy yields a Kakeya set;
the optimal rabbit strategy is based on a discretized Cauchy random walk,
and it yields a Kakeya set K consisting of 4n triangles, that has minimal
area among such sets (the area of K is of order 1/log(n)). Passing to
the scaling limit yields a simple construction of a random Kakeya set
with zero area from two Brownian motions. (Joint work with Y.
Babichenko, R. Peretz, P. Sousi and P. Winkler).

**Tutorial Seminar: ** What is the mixing time for random walk on a graph?

*Abstract:* Consider a simple random walk on a finite graph. The mixing time
is the
time it takes the walk to reach a position that is approximately
independent of the
starting point; it has been studied intensively by combinatorialists,
computer
scientists and probabilists; the mixing time arises in statistical physics
as well.
Applications of mixing times range from random sampling and card
shuffling, to
understanding convergence to equilibrium in the Ising model. It is closely
related
to expansion and eigenvalues. Besides introducing this topic, I will
also describe
the open problem of understanding which random walks exhibit "cutoff", a
sharp
transition to stationarity first discovered by Diaconis, Shashahani and
Aldous in
the early 1980s but still mysterious.

* Title: * ** Convergence to equilibrium for nonreversible diffusions. **

*Abstract:* The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. Powerful techniques from analysis and PDEs, such as spectral theory and functional inequalities (e.g. logarithmic Sobolev inequalities) can be used in order to study convergence to equilibrium. Quite often, the diffusion processes that appear in applications are degenerate (in the sense that noise acts directly to only some of the degrees of freedom of the system) and/or nonreversible. The study of convergence to equilibrium for such systems requires the study of non-selfadjoint, possibly non-uniformly elliptic, second order differential operators. In this talk we show how the recently developed theory of hypocoercivity can be used to prove exponentially fast convergence to equilibrium for such diffusion processes. Furthermore, we will show how the addition of a nonreversible perturbation to a reversible diffusion can speed up convergence to equilibrium. This is joint work with M. Ottobre, K. Pravda-Starov, T. Lelievre and F. Nier.

This is a special event. Billingsley Lectures on Probability in honor of Professor Billingsley.

*Title:* ** Random Walk with Reinforcement**

*Abstract:* Picture a triangle, with vertices labeled A, B, C. A random walker starts at A and chooses a random nearest neighbor. At each stage, the walker adds 1 to the weight of each crossed edge and chooses the next step with probability proportional to the current edge weights. The question is 'what happens?'. This simple problem leads into interesting corners: to Bayesian analysis of the transition mechanism of Markov chains (and protein folding) and to the hyperbolic sigma model of statistical physics. Work of (and with) Billingsley, Baccalado, Freedman, Tarres, and Sabot will be reviewed.

*Title:* ** Exchangeable random measures**

*Abstract:* Classical theorems of de Finetti, Aldous-Hoover and Kallenberg describe the structure of exchangeable probability measures on spaces of sequences or arrays. Similarly, one can add an extra layer of randomness, and ask after exchangeable random measures on these spaces. It turns out that those classical theorems, coupled with an abstract version of the `replica trick' from statistical physics, give a structure theorem for these random measures also. This leads to a new proof of the Dovbysh-Sudakov Theorem describing exchangeable positive semi-definite matrices.

*Title:* **Maximum independent sets in random d-regular graphs**

*Abstract:* Satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. These problems also arise naturally in combinatorics, in the study of sparse random graphs. While the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. In this context we study the size of maximum independent sets in random d-regular graphs. We show that for d exceeding a constant d(0), there exist explicit constants A, C depending on d such that the maximum size has constant fluctuations around A*n-C*(log n) establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. As an application of our method we also prove an explicit satisfiability threshold in random regular k-NAE-SAT.
This is joint work with Jian Ding and Allan Sly.

*Title: ***Scaling limit of the abelian sandpile**

*Abstract*: Which functions of two real variables can be expressed as
limits of superharmonic functions from (1/*n*)Z^{2} to (1/*n*^{2})Z? I'll
discuss joint work with Wesley Pegden and Charles Smart on the case of
quadratic functions, where this question has a surprising and
beautiful answer: the maximal such quadratics are classified by the
circles in a certain Apollonian circle packing of the plane. I'll also
explain where the question came from (the title is a hint!).

*Title: ***The relaxation of a family of broken bond crystal surface models**

*Abstract*: We study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation (PDE) limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational investigations of a number of interesting qualitative features of the large scale behavior of the models.

*Title: ***Random polymers and last passage percolation: variational formulas, Busemann functions, geodesics, and other stories**

*Abstract*: We give variational formulas for random polymer models, both in the positive- and zero-temperature cases. We solve these formulas in the oriented two-dimensional zero-temperature case. The solution comes via proving almost-sure existence of the so-called Busemann functions. We then use these results to prove existence, uniqueness, and coalescence of semi-infinite directional geodesics, for exposed points of differentiability of the limiting shape function.

*Title:* ** The scaling limit of simple triangulations and quadrangulations **

*Abstract:* A graph is simple if it contains no loops or multiple edges. We
establish Gromov--Hausdorff convergence of large uniformly random simple
triangulations and quadrangulations to the Brownian map, answering a
question of Le Gall (2011). In proving the preceding fact, we
introduce a labelling
function for the vertices of the triangulation. Under this labelling,
distances to a distinguished point are essentially given by vertex labels,
with an error given by the winding number of an associated closed loop in
the map. The appearance of a winding number suggests that a discrete
complex-analytic approach to the study of random triangulations may lead to
further discoveries.
Joint work with Marie Albenque.

* Title * ** Law of the extremes for the two-dimensional discrete Gaussian Free Field **

*Abstract: *
A two-dimensional discrete Gaussian Free Field (DGFF) is a centered Gaussian process over a finite subset
(say, a square) of the square lattice with covariance given by the Green function of the simple random walk
killed upon exit from this set. Recently, much effort has gone to the study of the concentration properties
and tail estimates for the maximum of DGFF. In my talk I will address the limiting extreme-order statistics of
DGFF as the square-size tends to infinity. In particular, I will show that for any sequence of squares along
which the centered maximum converges in law, the (centered) extreme process converges in law to a
randomly-shifted Gumbel Poisson point process which is decorated, independently around each point, by a random
collection of auxiliary points. If there is any time left, I will review what we know and/or believe about the
law of the random shift. This talk is based on joint work with Oren Louidor (UCLA).

*Title:* ** The Virasoro algebra and discrete Gaussian free field**

*Abstract:* The Virasoro algebra is an infinite dimensional Lie algebra
that plays an important role in the Conformal Field Theory (CFT)
methods employed by physicists to describe and study conformally
invariant scaling limits of planar critical lattice models from
statistical physics. Despite much progress in the last decade, it
seems fair to say that from a mathematical perspective many aspects of
the connections between discrete model and continuum limit CFT remain
somewhat mysterious. In the talk I will discuss recent joint work with
C. Hongler and K. Kytola concerning the discrete Gaussian free field
on a square grid. I will explain how for this model discrete complex
analysis can be used to construct explicit (exact) representations of
the Virasoro algebra of central charge 1 directly on the discrete
level.

*Title:* ** Markov type and the multi-scale geometry of metric spaces**

*Abstract:* The behavior of random walks on metric spaces can sometimes be understood by embedding such a walk into a nicer space (e.g. a Hilbert space) where
the geometry is more readily approachable. This beautiful theme has seen a number
of geometric and probabilistic applications. We offer a new twist on this study by showing that one can employ mappings that
are significantly weaker than bi-Lipschitz. This is used to answer
questions of Naor, Peres, Schramm, and Sheffield (2004) by proving that planar graph
metrics and doubling metrics have Markov type 2. The main new technical idea is that
martingales are significantly worse at aiming than one might at first expect.
Joint work with Jian Ding and Yuval Peres.

*Title:* ** The Renormalization Group for Disordered Systems**

*Abstract:* We investigate the Renormalization Group (RG) approach in finite- dimensional glassy systems, whose critical features are still not well-established, or simply unknown. We focus on spin and structural-glass models built on hierarchical lattices, which are the simplest non-mean-field systems where the RG framework emerges in a natural way. The resulting critical properties shed light on the critical behavior of spin and structural glasses beyond mean field, and suggest future directions for understanding the criticality of more realistic glassy systems.

*Title:* **Geodesics and Direction in 2d First-Passage
Percolation**

*Abstract:* I will discuss geodesics in first-passage percolation, a
model for fluid flow in a random medium. There are numerous conjectures
about the existence, coalescence, and asymptotic direction of infinite
geodesics under the model's random metric. C. Newman and collaborators
have proved some of these under strong assumptions. I will explain recent
results with Michael Damron which develop a framework for addressing these
questions; this framework allows us to prove versions of Newman's results
under minimal assumptions.

*Title:* ** Gaussian Free Field fluctuations for general-beta random
matrix ensembles. **

*Abstract:*
It is now known that the asymptotic fluctuations of the height function of
uniformly random lozenge tilings of planar domains (equivalently, stepped
surfaces in 3d space) are governed by the Gaussian Free Field (GFF), which
is a 2d analogue of the Brownian motion. On the other hand, in certain
limit regimes such tilings converge to various random matrix ensembles
corresponding to beta=2. This makes one wonder whether GFF should also
somehow arise in general-beta random matrix ensembles. I will explain that
this is indeed true and the asymptotics of fluctuations of classical
general-beta random matrix ensembles is governed by GFF. This is joint
work
with A.Borodin.

*Title:* ** About heavy tailed random matrices. **

*Abstract:*We investigate the behaviour of matrices which do not
belong to the universality class of Wigner matrices because their entries
have heavy tails.

*Title:* ** Chaos problem in mean field spin glasses **

*Abstract:* The main objective in spin glasses from the physical perspective is to understand the strange magnetic properties of certain alloys. Yet the models invented to explain the observed phenomena are also of a rather fundamental nature in mathematics. In this talk we will first introduce the famous Sherrington-Kirkpatrick model as well as some known results about this model such as the Parisi formula and the limiting behavior of the Gibbs measure. Next, we will discuss the problems of chaos in the mixed p-spin models and present mathematically rigorous results including disorder, external field, and temperature chaos.

*Abstract:* This seminar was canceled. It will be rescheduled.

*Title:* **The 2D SOS Model **

*Abstract:* We present new results on the (2+1)-dimensional Solid-On-Solid model at low temperatures. Bricmont, El-Mellouki and Froelich (1986) showed that in the presence of a floor there is an entropic repulsion phenomenon, lifting the surface to a height which is logarithmic in the side of the box. We refine this and establish the typical height of the SOS surface is precisely the floor of [1/(4\beta)\log n], where n is the side-length of the box and \beta is the inverse-temperature. We determine the asymptotic shape of the top plateau and show that its boundary fluctuation are n^{1/3+o(1)}. Based on joint works with Pietro Caputo, Eyal Lubetzky, Fabio Martinelli and Fabio Toninelli.

*Title:* ** Spiking the random matrix hard edge. **

*Abstract:* The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

*Title:* ** Directed polymers and the stochastic heat equation **

*Abstract:* We show how some properties of the solutions of the Stochastic Heat Equation (SHE) can be derived from directed polymers in random environment. In particular, we show:
* A new proof of the positivity of the solutions of the SHE
* Improved bounds on the negative moments of the SHE
* Results on the fluctuations of the log of the SHE in equilibrium, namely, the Cole-Hopf solution of the KPZ equation (if time allows).

*Title:* ** Second-order Boltzmann-Gibbs principle and applications **

*Abstract:* The celebrated Botzmann-Gibbs principle introduced by Rost in the 80's roughly says the following. For stochastic systems with one or more conservation laws, fluctuations of the non-conserved quantities are faster than fluctuations of the conserved quantities. Therefore, in the right space-time window, the space-time fluctuations of a given observable are asymptotically equivalent to a linear functional of the conserved quantities. In one dimension, we prove two generalizations of this principle: a non-linear (or second-order) and
a local version of it. This result opens a way to show convergence of
fluctuations for non-linear models, like the ones on the fashionable
KPZ universality class. As a corollary, we prove new convergence
results for various observables of the asymmetric exclusion process,
given in terms of solutions of the KPZ equation.
Joint work with Patricia Gonçalves.

*Title:* ** SLE curves and natural parametrization **

*Title:* ** Robust Gaussian noise stability **

*Abstract:* Given two Gaussian vectors that are positively correlated, what is the
probability that they both land in some fixed set A? Borell proved that
this probability is maximized (over sets A with a given volume) when A is a
half-space. We will give a new and simple proof of this fact, which also
gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost maximize this probability must be close to half-spaces.

*Title:* ** Spiking the random matrix hard edge. **

*Abstract:* The largest eigenvalue of a finite rank perturbation of a random hermitian matrix is known to exhibit a phase transition (in the infinite dimensional limit). If the perturbation is small one sees the famous Tracy-Widom law, while a large perturbation results in a Gaussian fluctuation. In between there exists is a scaling window about a critical perturbation value leading to a separate family of limit laws. This basic discovery is due to Baik, Ben Arous, and Peche. More recently Bloemendal and Virag have shown this picture persists in the context of the general beta ensembles, giving new formulations of the critical limit laws . Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to distributions. (Joint work with Jose Ramirez.)

*Title:* ** Extreme values for random processes of tree structures **

*Abstract:* The main theme of this talk is that studying implicit tree structures
of random processes is of significance in understanding their extreme values. I will illustrate this by several examples including cover times for random walks, maxima for two-dimensional discrete Gaussian free fields, and stochastic distance models. Our main results include
(1) An approximation of the cover time on any graph up to a multiplicative constant by the maximum of the Gaussian free field, which yields a deterministic polynomial-time approximation algorithm for the cover time (D.-Lee-Peres 2010); the asymptotics for the cover time on a bounded-degree graph by the maximum of the GFF (D. 2011); a bound on the cover time fluctuations on the 2D lattice (D. 2011).
(2) Exponential and doubly exponential tails for the maximum of the 2D GFF (D. 2011); some results on the extreme process of the 2D GFF (D.-Zeitouni, in preparation).
(3) Critical and near-critical behavior for the mean-field stochastic distance model (D. 2011).

*Title:* ** Imaginary Geometry and the Gaussian Free Field **

*Abstract:* The Schramm-Loewner evolution (SLE) is the canonical model of a non-crossing conformally invariant random curve, introduced by Oded Schramm in 1999 as a candidate for the scaling limit of loop erased random walk and the interfaces in critical percolation. The development of SLE has been one of the most exciting areas in probability theory over the last decade because Schramm's curves have now been shown to arise as the scaling limit of the interfaces of a number of different discrete models from statistical physics. In this talk, I will describe how SLE curves can be realized as the flow lines of a random vector field generated by the Gaussian free field, the two-time-dimensional analog of Brownian motion. I will also explain how this perspective can be used to prove several new results regarding the sample path behavior of SLE, in particular reversibility for kappa in (4,8).
Based on joint works with Scott Sheffied.

*Title:* **Directed random polymers and Macdonald processes**

*Abstract:* The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

*Title:* **Path properties of the Schramm-Loewner Evolution.**

*Title:* **Extrema of branching Brownian motion**

*Abstract:* Branching Brownian motion (BBM) on the real line is a particle system where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The statistics of extremal particles of BBM in the limit of large time are of interest for physicists and probabilists since BBM constitutes a borderline case, among Gaussian processes, where correlations affect the statistics. In this talk, I will start by reviewing results on the law of the maximum of BBM (the rightmost particle), and present new results on the joint distribution of particles close to the maximum. In particular, I will show how the approach can be used to prove ergodicity of the particle system. If time permits, I will explain how the program for BBM lays out a road map to understand extrema of log-correlated Gaussian fields such as the 2D Gaussian free field. This is joint work with A. Bovier and N. Kistler.

This is a special event. Billingsley Lectures on Probability in honor of Patrick Billingsley

*Abstract:* See it here.

*Abstract:* See it here.

*Title:* ** Landscape of random functions in many dimensions via Random Matrix Theory.**

*Abstract:* How many critical values a typical Morse function have on a high
dimensional manifold? Could we say anything about the topology of its level sets? In
this talk I will survey a joint work with Gerard Ben Arous and Jiri Cerny that
addresses these questions in a particular but fundamental example. We investigate
the landscape of a general Gaussian random smooth function on the N-dimensional
sphere. These corresponds to Hamiltonians of well-known models of statistical
physics, i.e spherical spin glasses. Using the classical Kac-Rice formula, this
counting boils down to a problem in Random Matrix Theory. This allows us to show an
interesting picture for the complexity of these random Hamiltonians, for the bottom
of the energy landscape, and in particular a strong correlation between the index
and the critical value. We also propose a new invariant for the possible transition
between the so-called 1-step replica symmetry breaking and a Full Replica symmetry
breaking scheme and show how the complexity function is related to the Parisi
functional.

*Title:* **Finite-rank deformations of Wigner matrices.**

*Abstract:* The spectral statistics of large Wigner matrices are by now
well-understood. They exhibit the striking phenomenon of universality:
under very general assumptions on the matrix entries, the limiting
spectral statistics coincide with those of a Gaussian matrix ensemble.
I shall talk about Wigner matrices that have been perturbed by a
finite-rank matrix. By Weyl's interlacing inequalities, this
perturbation does not affect the large-scale statistics of the
spectrum. However, it may affect eigenvalues near the spectral edge,
causing them to break free from the bulk spectrum. In a series of
seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006)
established a sharp phase transition in the statistics of the extremal
eigenvalues of perturbed Gaussian matrices. At the BBP transition, an
eigenvalue detaches itself from the bulk and becomes an outlier.
I shall report on recent joint work with Jun Yin. We consider an NxN
Wigner matrix H perturbed by an arbitrary deterministic finite-rank
matrix A. We allow the eigenvalues of A to depend on N. Under optimal
(up to factors of log N) conditions on the eigenvalues of A, we
identify the limiting distribution of the outliers. We also prove that
the remaining eigenvalues "stick" to eigenvalues of H, thus
establishing the edge universality of H + A. On the other hand, our
results show that the distribution of the outliers is not universal,
but depends on the distribution of H and on the geometry of the
eigenvectors of A. As the outliers approach the bulk spectrum, this
dependence is washed out and the distribution of the outliers becomes
universal.

Jonathan Mattingly - *Duke University*

*Title:* **A Menagerie of Stochastic Stabilization**

*Abstract:* A basic problem for a stochastic system is to show that it possesses a unique steady state which dictates the long term statistics of the system. Sometimes the existence of such a measure is the difficult part. One needs control of the excursions away from the systems typical scale. As in deterministic system, one popular method is the construction of a Lyapunov Function. In the stochastic setting there lack of systematic methods to construct a Lyapunov Function when the interplay between the deterministic dynamics and stochastic dynamics are important for stabilization. I will give some modest steps in this direction which apply to a number of cases. In particular I will show a system where an explosive deterministic system is stabilized by the addition of noise and examples of physical systems where it is not clear how the deterministic system absorbs the stochastic excitation with out blowing up.

*Title:* ** From random interlacements to coordinate and infinite cylinder
percolation**

*Abstract:* During the talk I will focus on the connectivity properties of
three models with long (infinite)
range dependencies: Random Interlacements, percolation of the vacant set
in infinite
rod model and Coordinate percolation. The latter model have polynomial
decay in sub-critical and super-critical regime in dimension 3.
I will explain the nature of this phenomenon and why it is difficult to
handle these models technically. In the second half
of the talk I will present key ideas of the multi-scale analysis which
allows to reach some conclusions. At the end I will discuss
applications and several open problems.

*Title:* ** Complete matchings and random matrix theory**

*Abstract:* Over the last decade or so, it has been found that the distributions that
first appeared in random matrix theory describe several objects in probability and
combinatorics which do not come from matrix at all. We consider one such example
from the so-called maximal crossing and nesting of random complete matchings of
integers. We also discuss related non-intersecting process. This is a joint work
with Bob Jenkins.

*Title:* **A simplified proof of the relation between scaling exponents in
first-passage percolation**

*Abstract:* In first passage percolation, we place i.i.d. non-negative
weights on the nearest-neighbor edges of Z^d and study the induced
random metric. A long-standing conjecture gives a relation between two
"scaling exponents": one describes the variance of the distance
between two points and the other describes the transversal
fluctuations of optimizing paths between the same points. This is
sometimes referred to as the "KPZ relation." In a recent breakthrough
work, Sourav Chatterjee proved this conjecture using a strong
definition of the exponents. I will discuss work I just completed with
Tuca Auffinger, in which we introduce a new and intuitive idea that
replaces Chatterjee's main argument and gives an alternative proof of
the relation. One advantage of our argument is that it does not
require a certain non-trivial technical assumption of Chatterjee on
the weight distribution.

Ofer Zeitouni - *University of Minnessota *

*Title:* **Traveling waves, branching random walks, and the Gaussian free field**

*Abstract:* I will discuss several aspects of Branching random walks and their relation with the KPP equation on the one hand, and the maximum of certain (two dimensional) Gaussian fields on the other. I will not assume any knowledge about either of these terms.

*Title:* ** The parafermionic observable in Schramm-Loewner Evolutions **

*Abstract:* In recent years, work by Stanislav Smirnov and his co-authors has greatly advanced our understanding of discrete stochastic processes, such as self-avoiding walk and the Ising model, via the use of a tool known as the parafermionic observable. Much of that work has been done in order to show convergence of these models to Schramm-Loewner Evolutions (SLE) in the scaling limit, although very little work has been done on what the parafermionic observable is in SLE itself. In this talk I will introduce the parafermionic observable, and then discuss one possible generalization to the continuous setting. I will then briefly introduce SLE and compute its parafermionic observable, ending with a couple of open questions.

*Title:* ** The contact process on the complete graph with random, vertex-dependent infection rates.
**

*Abstract:* The contact process is an interacting particle system that is a very simple model for the spread of an infection or disease on a network. Traditionally, the contact process was studied on homogeneous graphs such as the integer lattice or regular trees. However, due to the non-homogeneous structure of many real-world networks, there is currently interest in studying interacting particle systems in non-homogeneous graphs and environments. In this talk, I consider the contact process on the complete graph, where the vertices are assigned (random) weights and the infection rate between two vertices is proportional to the product of their weights. This set-up allows for some interesting analysis of the process and detailed calculations of phase transitions and critical exponents.

*Title:* ** Universality for beta-ensembles. **

* Abstract: * Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis is for ensembles of large but finite dimensional matrices.
Spectacular progress was done in the past decade to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4.
I will report on a joint work with L. Erd\"os and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature. The involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

Friday, Oct. 8, Fredrik Johansson Viklund, Columbia U.

Friday, Oct. 15, Midwest Probability Colloquium at Northwestern

Friday, Oct. 29, Tom Alberts, U. of Toronto,
Convergence of Loop-Erased Random Walk to SLE(2) in the
Natural Time Parameterization

I will discuss work in progress with Michael Kozdron and Robert Masson on the convergence of the two-dimensional loop-erased random walk process to SLE(2), with the time parameterization of the curves taken into account. This is a strengthening of the original Lawler, Schramm, and Werner result which was only for curves modulo a reparameterization. The ultimate goal is to show that the limiting curve is SLE(2) with the very specific natural time parameterization that was recently introduced in Lawler and Sheffield, and further studied in Lawler and Zhou. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others.

Friday, Dec. 3, Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK
Ising model

For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality. In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - including some new ones - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent. This is joint work with H. Duminil-Copin and C. Hongler.