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) and an affiliate member of the statistics department

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 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

• Jinwoo Sung
• Charles Devlin
• Andres Contreras Hip
• Mark Olson
• Yuyang Feng
• Brin Harper
• Deven Mithal
• Leonardo Bonanno
• Yasmeen Shaw

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

• Liouville Brownian motion and quantum cones in dimension d > 2 (with F. Bertacco). arXiv:2501.15936
• Percolation of thick points of the log-correlated Gaussian field in high dimensions (with J. Ding and Z. Zhuang). arXiv:2410.22075
• 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
• A support theorem for exponential metrics of log-correlated Gaussian fields in arbitrary dimension (with A. Contreras Hip). arXiv:2305.15588

Relationships between supercritical LQG, SLE, and random planar maps

• Inverting the operation of conditioning a branching process on extinction (with J. Liu). arXiv:2411.06301
• Area measures and branched polymers in supercritical Liouville quantum gravity (with M. Bhatia and J. Sung). arXiv:2410.12693
• Supercritical Liouville quantum gravity and CLE(4) (with M. Ang). arXiv:2308.11832
• Critical Liouville quantum gravity and CLE(4) (with M. Ang). Annales de l'Institut Henri Poincarè. arXiv:2308.11835

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

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). Proceedings of the London Mathematical Society. arXiv:2208.11795

Meanders and permutations

• Reconstructing SLE-decorated Liouville quantum gravity surfaces from random permutons (with J. Borga). arXiv:2409.15494
• 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). Journal of the European Mathematical Society. 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

• 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). Annals of Probability. 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). Acta Mathematica Sinica, English Series. 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

• Apr. 2025. Complex faces conference in Venice.
• 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)