Ewain Gwynne

I am a professor of mathematics at the University of Chicago. I am also a member of the Committee on Computational and Applied Mathematics (CCAM).

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 geometric objects 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 an associate editor of Probability and Mathematical Physics.

Contact Information

Office: Ryerson 360D

Email: ewain at uchicago.edu

Here is a link to my CV: CV

PhD students advised (at least in part) by me

Articles

Expository works

Introduction to the Liouville quantum gravity metric (with J. Ding and J. Dubedat). Proceedings of the ICM, 2022. 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

Curvature in random geometry

Gaussian curvature on random planar maps and Liouville quantum gravity (with A. Contreras Hip). arXiv:2406.08674

Random geometry in dimension at least three

Random walk on sphere packings and Delaunay triangulations in arbitrary dimension (with A. Bou-Rabee). arXiv:2405.11673
Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension (with J. Ding and Z. Zhuang). arXiv:2310.03996

Relationships between Liouville quantum gravity and SLE with mismatched parameter values

Cutting γ-Liouville quantum gravity by Schramm-Loewner evolution for κ not in {γ^2,16/γ^2} (with M. Ang). arXiv:2310.11455

Relationships between supercritical LQG, SLE, and random planar maps

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

Harmonic balls and IDLA

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

Meanders and permutations

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

Loewner evolution with complex driving function

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

Critical and supercritical Liouville quantum gravity metric

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). Frontiers in Mathematics. 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

Subcritical Liouville quantum gravity metric

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). Annals of Probability. 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

Distances in random planar maps and Liouville quantum gravity

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)

Random walk/conformal embeddings of random planar maps

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

Percolation on random planar maps

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 SLE₆ 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 SLE₆ explorations of a quantum disk (with J. Miller). Electronic Journal of Probability. arxiv:1701.05172

Self-avoiding walk on random planar maps

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)

Fractal properties of SLE/LQG

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)

Peanosphere convergence/mating-of-trees bijections

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

First passage percolation

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

Undergraduate papers

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)

Talks

Mar. 2025. University of Wisconsin Madison probability seminar.
Dec. 2024. UIUC colloquium.
Sep. 2024. MIT colloquium.
Sep. 2024. MIT probability seminar.
Sep. 2024. Purdue probability seminar.
Jul. 2024. Recent Progress in Stochastic Analysis and its Applications, Loyola University.
Mar. 2024. UChicago computational and applied math colloquium.
Feb. 2024. Michigan State colloquium.
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)