I am an associate professor of mathematics at the University of Chicago.

Prior to coming to Chicago I was a postdoc at the University of Cambridge. I got my Ph.D from MIT in 2018, advised by Scott Sheffield. Previously, I was an undergraduate student at Northwestern University.

I am interested in probability theory, particularly random curves and surfaces which arise in statistical mechanics. Topics I have worked on include Schramm-Loewner evolution, Liouville quantum gravity, random planar maps, random permutations, random walk in random environment, and various random growth processes.

Here is a non-technical summary of my research aimed at an audience with some scientific background, not necessarily in math.

I am a co-organizer of the workshop Two-Dimensional Random Geometry which will take place at IMSI in Chicago, July 8-12 2024.

I am a co-organizer of the online seminar Random Geometry and Statistical Physics.

I am an associate editor of Probability and Mathematical Physics.

*Office:* Ryerson 360D

*Email:* ewain *at* uchicago.edu

Here is a link to my CV: CV

- Introduction to the Liouville quantum gravity metric (with J. Ding and J. Dubedat). arXiv:2109.01252
- Mating of trees for random planar maps and Liouville quantum gravity: a survey (with N. Holden and X. Sun).
*Panoramas et Syntheses*. arXiv:1910.04713 - Random surfaces and Liouville quantum gravity.
*Notices of the American Mathematical Society*. arXiv:1908.05573

- Supercritical Liouville quantum gravity and CLE(4) (with M. Ang). arXiv:2308.11832
- Critical Liouville quantum gravity and CLE(4) (with M. Ang). arXiv:2308.11835

- Internal DLA on mated-CRT maps (with A. Bou-Rabee). arXiv:2211.04891
- Harmonic balls in Liouville quantum gravity (with A. Bou-Rabee). arXiv:2208.11795

- Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons (with J. Borga and W. Da Silva). arXiv:2303.17030
- On the geometry of uniform meandric systems (with J. Borga and M. Park). arXiv:2212.00534
- Permutons, meanders, and SLE-decorated Liouville quantum gravity (with J. Borga and X. Sun). arXiv:2207.02319

- Loewner evolution driven by complex Brownian motion (with J. Pfeffer, simulations by M. Park).
*Annals of Probability*. arXiv:2203.07313

- Uniqueness of the critical and supercritical Liouville quantum gravity metrics (with J. Ding).
*Proceedings of the London Mathematical Society.*arXiv:2110.00177 - The critical Liouville quantum gravity metric induces the Euclidean topology (with J. Ding). arXiv:2108.12067
- Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity (with J. Ding).
*Science China Mathematics.*arXiv:2108.12060 - Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric (with J. Ding).
*Probability and Mathematical Physics*. arXiv:2104.06502 - Tightness of supercritical Liouville first passage percolation (with J. Ding).
*Journal of the European Mathematical Society.*arXiv:2005.13576 - The distance exponent for Liouville first passage percolation is positive (with J. Ding and A. Sepulveda).
*Probability Theory and Related Fields*. arXiv:2005.13570 - Liouville quantum gravity with matter central charge in (1,25): a probabilistic approach (with N. Holden, J. Pfeffer, and G. Remy).
*Communications in Mathematical Physics*. arXiv:1903.09111

- A support theorem for Liouville quantum gravity (with A. Contreras Hip). arXiv:2305.15588
- The Minkowski content measure for the Liouville quantum gravity metric (with J. Sung). arXiv:2211.04701
- Geodesic networks in Liouville quantum gravity surfaces.
*Probability and Mathematical Physics*. arXiv:2010.11260 - Geodesics and metric ball boundaries in Liouville quantum gravity (with J. Pfeffer and S. Sheffield).
*Probability Theory and Related Fields*. arXiv:2010.07889 - The dimension of the boundary of a Liouville quantum gravity metric ball.
*Communications in Mathematical Physics*. arXiv:1909.08588 - KPZ formulas for the Liouville quantum gravity metric (with J. Pfeffer).
*Transactions of the American Mathematical Society*. arXiv:1905.11790 - Conformal covariance of the Liouville quantum gravity metric for γ ∈ (0,2) (with J. Miller).
*Annales de l'Institut Henri Poincarè*. arXiv:1905.00384 - Existence and uniqueness of the Liouville quantum gravity metric for γ ∈ (0,2) (with J. Miller).
*Inventiones Mathematicae*. arXiv:1905.00383 - Confluence of geodesics in Liouville quantum gravity for γ ∈ (0,2) (with J. Miller).
*Annals of Probability*. arXiv:1905.00381 - Weak LQG metrics and Liouville first passage percolation (with J. Dubedat, H. Falconet, J. Pfeffer, and X. Sun).
*Probability theory and related fields*. arXiv:1905.00380 - Local metrics of the Gaussian free field (with J. Miller).
*Annales de l'Institut Fourier*. arXiv:1905.00379

- Bounds for distances and geodesic dimension in Liouville first passage percolation (with J. Pfeffer).
*Electronic Communications in Probability*. arXiv:1903.09561 - External diffusion limited aggregation on a spanning-tree-weighted random planar map (with J. Pfeffer).
*Annals of Probability*. arXiv:1901.06860 - The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds (with J. Ding).
*Communications in Mathematical Physics*. arXiv:1807.01072 - A mating-of-trees approach for graph distances in random planar maps (with N. Holden and X. Sun).
*Probability Theory and Related Fields*. arXiv:1711.00723 - A distance exponent for Liouville quantum gravity (with N. Holden and X. Sun).
*Probability Theory and Related Fields*. arXiv:1606.01214 (journal)

- Scaling limits of planar maps under the Smith embedding (with F. Bertacco and S. Sheffield). arXiv:2306.02988
- Random walks on mated-CRT planar maps and Liouville Brownian motion (with N. Berestycki).
*Communications in Mathematical Physics*. arXiv:2003.10320 - The Tutte embedding of the Poisson-Voronoi tessellation of the Brownian disk converges to √ 8/3 -Liouville quantum gravity (with J. Miller and S. Sheffield).
*Communications in Mathematical Physics*. arXiv:1809.02091 - Harmonic functions on mated-CRT maps (with J. Miller and S. Sheffield).
*Electronic Journal of Probability*. arXiv:1807.07511 - An invariance principle for ergodic scale-free random environments (with J. Miller and S. Sheffield).
*Acta Mathematica*. arXiv:1807.07515 - Anomalous diffusion of random walk on random planar maps (with T. Hutchcroft).
*Probability theory and related fields*. arXiv:1807.01512 - Random walk on random planar maps: spectral dimension, resistance, and displacement (with J. Miller).
*Annals of Probability*. arxiv:1711.00836 - The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity (with J. Miller and S. Sheffield).
*Annals of Probability*. arxiv:1705.11161

- Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense (with N. Holden and X. Sun).
*Electronic Journal of Probability.*arXiv:1905.06757 - Convergence of percolation on uniform quadrangulations with boundary to SLE
_{6}on √ 8/3 -Liouville quantum gravity (with J. Miller).*Asterisque*. arxiv:1701.05175 - Characterizations of SLE
_{κ}for κ ∈ (4,8) on Liouville quantum gravity (with J. Miller).*Asterisque*. arxiv:1701.05174 - Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk (with J. Miller).
*Annales de l'Institut Henri Poincarè*. arxiv:1701.05173 - Chordal SLE
_{6}explorations of a quantum disk (with J. Miller).*Electronic Journal of Probability*. arxiv:1701.05172

- Convergence of the self-avoiding walk on random quadrangulations to SLE
_{8/3}on √ 8/3 -Liouville quantum gravity (with J. Miller).*Annals de l'ENS*. arxiv:1608.00956 - Metric gluing of Brownian and √ 8/3 -Liouville quantum gravity surfaces (with J. Miller).
*Annals of Probability*. arxiv:1608.00955 - Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology (with J. Miller).
*Electronic Journal of Probability*. arxiv:1608.00954 (journal)

- Liouville quantum gravity surfaces with boundary as matings of trees (with M. Ang).
*Annales de l'Instut Henri Poincare*. arXiv:1903.09120 - Conformal invariance of CLE
_{κ}on the Riemann sphere for κ ∈ (4,8) (with J. Miller and W. Qian).*International Math Research Notices*. arxiv:1811.00514 - Connectivity properties of the adjacency graph of SLE
_{κ}bubbles for κ ∈ (4,8) (with J. Pfeffer).*Annals of Probability*. arxiv:1803.04923 - Dimension transformation formula for conformal maps into the complement of an SLE curve (with N. Holden and J. Miller).
*Probability theory and related fields*. arxiv:1603.05161 - An almost sure KPZ relation for SLE and Brownian motion (with N. Holden and J. Miller).
*Annals of Probability*. arxiv:1512.01223 - Brownian motion correlation in the Peanosphere for κ > 8 (with N. Holden, J. Miller, and X. Sun).
*Annales de l'Institut Henri Poincarè*. arxiv:1510.04687 (journal) - Almost sure multifractal spectrum of SLE (with J. Miller and X. Sun).
*Duke Mathematical Journal*. arxiv:1412.8764 (journal)

- Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for κ > 8 (with
A. Kassel,
J. Miller, and
D. Wilson).
*Communications in Mathematical Physics*. arxiv:1603.09722 (journal) - Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense (with N. Holden and X. Sun). arxiv:1603.01194
- Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map III: finite volume case (with X. Sun). arxiv:1510.06346
- Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map II: local estimates and empty reduced word exponent (with X. Sun).
*Electronic Journal of Probability*. arxiv:1505.03375 (journal) - Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times (with C. Mao and X. Sun).
*Annales de l'Institut Henri Poincarè*. arxiv:1502.00546

- Asymptotic behavior of the Eden model with positively homogeneous edge weights (with S. Bubeck). arxiv:1508.05140

- On Beckner's Inequality for Gaussian Measures (with E. Hsu).
*Elemente der Mathematik*. (journal) - Functional Inequalities for Gaussian and Log-Concave Probability Measures. Undergraduate Thesis.
*Northwestern University Undergraduate Research Journal*. Adviser: Elton Hsu. (journal) - On a Quaternionic Analogue of the Cross Ratio (with M. Libine)
*Advances in Applied Clifford Algebras*. arxiv:1112.0612 - The Poisson Integral Formula and Representations of SU(1,1).
*Rose-Hulman Undergraduate Math Journal*. (journal)

- Mar. 2024. KPZ meets KPZ workshop, Fields Institute, Toronto.
- Aug. 2023. Mini course at Aalto University, Helsinki. Video recordings available here.
- Mar. 2023. CU Boulder probability seminar.
- Feb. 2023. NYU Courant probability seminar.
- Jan. 2023. UChicago physics colloquium.
- Oct. 2022. Institute for advanced study probability seminar.
- Sep. 2022.
*Universality in mathematical physics*mini school in Lyon (lecture notes. - Sep. 2022. Kendall award lecture, Royal Statistical Society meeting in Aberdeen.
- Jul. 2022. ICM sectional lecture (online). Joint talk with Jian Ding and Julien Dubedat.
- Jul. 2022.
*Probability and Mathematical Physics*ICM satellite conference in Helsinki (video). The talk is a solo version of my joint ICM sectional talk. - May 2022.
*Universality: Random Matrices, Random Geometry and SPDEs*conference at Oberwolfach. - Apr. 2022. UChicago math colloquium.
- Apr. 2022. Stanford probability seminar.
- Mar. 2022.
*The Analysis and Geometry of Random Spaces*workshop at MSRI. - Feb. 2022. Lyon probability seminar (online).
- Nov. 2021. Geneva Mathematical Physics seminar (online).
- Jan. 2021. Expository talk for Trinity college, Cambridge alumni (online).
- Dec. 2020. Oberseminar Stochastik at University of Bonn (online).
- Nov. 2020. University of Bristol probability seminar (online).
- Oct. 2020. AMS Sectional meeting (originally at Penn state, now online).
- Aug. 2020. Bernoulli-IMS One World Symposium, ``Models in Physics" session. (online)
- Apr. 2020. University of Bath probability seminar (online).
- Mar. 2020. Cambridge mathematical physics seminar.
- Jan. 2020. Wharton (UPenn) statistics seminar.
- Dec. 2019. Northwestern University colloquium.
- Dec. 2019. University of Chicago colloquium.
- Dec. 2019. University of Chicago probability seminar.
- Nov. 2019.
*Heat Kernels, Stochastic Processes and Functional Inequalities*workshop at Oberwolfach. - Oct. 2019. EPFL probability seminar.
- Sep. 2019. Princeton University probability seminar.
- Sep. 2019. Penn/Temple probability seminar.
- Jul. 2019. Stochastic Processes and their applications conference at Northwestern University.
- Jun. 2019. Probability and quantum field theory: discrete models, CFT, SLE and constructive aspects, conference in Porquerolle, France.
- May 2019. University of Cambridge probability seminar.
- Mar. 2019. Vienna probability seminar.
- Feb. 2019. University of Warwick probability seminar.
- Dec. 2018. Amir Dembo birthday conference, Stanford University.
- Oct. 2018. Imerial College London stochastic analysis seminar.
- Oct. 2018. Columbia University probability seminar.
- Jul. 2018. Random Geometry followup workshop at the Isaac Newton Institute. (video)
- Jun. 2018. IST Austria Summer School in Probability and Mathematical Physics.
- Mar. 2018. Stony Brook University analysis seminar.
- Feb. 2018. Brown University graduate student conference.
- Feb. 2018. Penn/Temple probability seminar.
- Dec. 2017. Tel Aviv university probability seminar.
- Oct. 2017. Oberwolfach seminar: Scaling limits of random planar maps and Liouville quantum gravity.
- Oct. 2017. Zurich graduate student probability seminar.
- Oct. 2017. Zurich probability seminar.
- Sep. 2017. Princeton University topics in probability seminar.
- May 2017. Stochastic Analysis: Geometry of Random Processes workshop at Oberwolfach.
- Apr. 2017. Brown discrete math seminar.
- Apr. 2017. AMS Sectional meeting at Indiana University.
- Mar. 2017. SLE, GFF, and LQG in NYC workshop at Columbia University.
- Feb. 2017. Cornell probability seminar.
- Jun. 2016. Recent developments in SLE conference at the Institut Mittag-Leffler.
- Feb. 2016. MIT probability seminar.
- Jan. 2016. University of Chicago probability seminar.
- Nov. 2015. Michigan State University probability seminar.
- Oct. 2015. Northwestern University analysis seminar.
- Aug. 2015.Microsoft Research, Redmond, WA. (video)
- Jan. 2015. Conformally invariant scaling limits conference at the Isaac Newton Institute. (video)