I am an associate professor at the University of Chicago. I also hold a Clay research fellowship and a junior research fellowship at Trinity college. 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, first passage percolation, and Brownian surfaces.
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 online seminar Random Geometry and Statistical Physics, which meets Thursdays at 10 CST.
Office: Ryerson 360D
Email: ewain at uchicago.edu
Here is a link to my CV: CV
The Liouville quantum gravity metric
- Geodesic networks in Liouville quantum gravity surfaces. arXiv:2010.11260
- Geodesics and metric ball boundaries in Liouville quantum gravity (with J. Pfeffer and S. Sheffield). 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
Liouville quantum gravity with matter central charge greater than 1
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). 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
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). arXiv:1905.06757
- Convergence of percolation on uniform quadrangulations with boundary to SLE6 on √ 8/3 -Liouville quantum gravity (with J. Miller). arxiv:1701.05175
- Characterizations of SLEκ for κ ∈ (4,8) on Liouville quantum gravity (with J. Miller). 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 SLE8/3 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
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
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
- 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)
- Jan. 2020. Expository talk for Trinity college, Cambridge alumni (online).
- Dec. 2020. UOberseminar 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)