##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability & Statistics

## Karl Liechty

#### De Paul University

## Tacnode processes, winding numbers, and Painleve II

##### Abstract:

I will discuss a model of nonintersecting Brownian bridges on the unit circle, which produces quite a few universal determinantal processes as scaling limits. I will focus on the tacnode process, in which two groups of particles meet at a single point in space-time before separating, and introduce a new version of the tacnode process in which a finite number of particles ``switch sides'' before the two groups separate. We call this new process the k-tacnode process, and it is defined by a kernel expressed in terms of a system of tau-functions for the Painleve II equation. Technically, our model of nonintersecting Brownian bridges on the unit circle is studied using a system of discrete orthogonal polynomials with a complex (non-Hermitian) weight, so I'll also discuss some of the analytical obstacles to that analysis. \noindent This is joint work with Dong Wang and Robert Buckingham

Host: Tianyi Zheng

### March 15, 2018

### 10:00 AM

### AP&M 6402

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