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 November 3rd 2017, Courant institute, Warren Weaver Hall, 512

- 9.30-10.30am
**Alisa Knizel (Columbia)** .
*Log-gases on a quadratic lattice via discrete loop equations
and q-boxed plane partitions *

We study a general class of log-gas ensembles on a quadratic lattice.
Using a variational principle we prove that the corresponding
empirical measures satisfy a law of large numbers and that their
global fluctuations are Gaussian with a universal covariance. We apply
our general results to analyze the asymptotic behavior of a q-boxed
plane partition model introduced by Borodin, Gorin and Rains. In
particular, we show that the global fluctuations of the height
function on a fixed slice are described by a one-dimensional section
of a pullback of the two-dimensional Gaussian free field.

Our approach is based on a q-analogue of the Schwinger-Dyson (or
loop) equations, which originate in the work of Nekrasov and his
collaborators, and extends the methods developed by Borodin, Gorin and
Guionnet to a quadratic lattice.

- 10.30-11am
**Coffee break**
* *

- 11am-12pm
**Leonid Koralov (Univ. Maryland)**
* Large Time Behavior of Randomly Perturbed Dynamical Systems *

We will discuss several asymptotic problems for randomly perturbed flows. One class of flows (with
regions where a strong flow creates a trapping mechanism) leads to a new class of boundary value problems with
non-standard boundary conditions. We will also discuss how large-deviation techniques can be used
to study the asymptotic behavior of solutions to quasi-linear parabolic equations with a small parameter at the
second order term and the long time behavior of the corresponding diffusion processes.

- 12-1pm
**Ben Landon (Harvard)**
* Local statistics of Dyson Brownian motion *.

Abstract: Dyson Brownian motion is a stochastic eigenvalue dynamics and the basis of the dynamical approach to random matrix theory. We review recent results on the local statistics of Dyson Brownian motion, as well as how these results are applied to prove the universality of general random matrix ensembles. Based on joint work with Z. Che, J. Huang, P. Sosoe and H.-T. Yau.