C O L U M B I A /
C O U R A N T
P R O B A B I L I T Y
S E M I N A R
S E R I E S

Joint Columbia-NYU probability talks, organized by probabilists from both institutions.

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Friday October 10th, Courant institute, Warren Weaver Hall, 512

- 9.30-10.20am
**Yuri Bakhtin** (Courant Institute).
*Scaling limits in diffusion exit problems.*.

In a recent paper with Andrzej Swiech, we were able to
describe a class of situations where a Gaussian scaling limit for
the exit point of conditioned diffusions holds, with the help of
Doob's h-transform and new gradient estimates for Hamilton--Jacobi
equations. I will talk about this result and about my earlier results
on exit problems including those on noisy heteroclinic
networks and neural dynamics models.

- 10.30-11.20am
**Paul Bourgade** (Courant Institute).
*Homogenization of the Dyson Brownian motion *.

I will explain a homogenization result for the Dyson Brownian motion, which gives microscopic statistics from mesoscopic ones. It implies in particular the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. This is joint work with L. Erdos, H.-T. Yau and J. Yin.

- 11.30-12.20am
**Ivan Corwin** (Columbia).
* Spectral theory and duality for interacting particle systems solvable by coordinate Bethe ansatz.*

We describe recent advances in Bethe ansatz and Markov dualities related to the q-Hahn stochastic particle system. Limits of the system or its spectral theory have applications in studying various systems such as ASEP, q-TASEP, XXZ spin chain, delta Bose gas, and the KPZ equation. This is based off of joint work with Alexei Borodin, Leonid Petrov, and Tomohiro Sasamoto.

- 12.20-1pm
**Lunch, tea and coffee, 13th floor**

- 1-1.50pm
**Eyal Lubetzky** (Courant Institute).
* Harmonic pinnacles in the Discrete Gaussian model*.

The 2D Discrete Gaussian model gives each height function $\eta : Z^2 \to Z$ a probability proportional to $\exp[-\beta H(\eta)]$, where $\beta$ is the inverse-temperature and $H(\eta)=\sum (\eta_x-\eta_y)^2$ sums over nearest-neighbor bonds. We consider the model at large fixed $\beta$, where it is flat unlike its continuous analog (the Gaussian Free Field).

We first establish that the maximum height in an $L\times L$ box with 0 boundary conditions concentrates on two integers $M,M+1$ with $M\sim [(2/\pi\beta)\log L \log\log L]^{1/2}$. The key is a large deviation estimate for the height at the origin in Z^2, dominated by ``harmonic pinnacles'', integer approximations of a harmonic variational problem. Second, in this model conditioned on $\eta \geq 0$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels $H,H+1$ where $H\sim M/\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and FrÃ¶hlich (1986). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices.

Joint work with Fabio Martinelli and Allan Sly.