# Probability Seminar

# Spring 2015

**Thursdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.

**
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**

## Thursday, January 28, Leonid Petrov, University of Virginia

Title: **The quantum integrable particle system on the line**

I will discuss the higher spin six vertex model - an interacting particle system on the discrete 1d line in the Kardar--Parisi--Zhang universality class. Observables of this system admit explicit contour integral expressions which degenerate to many known formulas of such type for other integrable systems on the line in the KPZ class, including stochastic six vertex model, ASEP, various [math]q[/math]-TASEPs, and associated zero range processes. The structure of the higher spin six vertex model (leading to contour integral formulas for observables) is based on Cauchy summation identities for certain symmetric rational functions, which in turn can be traced back to the sl2 Yang--Baxter equation. This framework allows to also include space and spin inhomogeneities into the picture, which leads to new particle systems with unusual phase transitions.

## Thursday, February 4, Inina Nenciu, UIC, Joint Probability and Analysis Seminar

Title: **On some concrete criteria for quantum and stochastic confinement**

Abstract: In this talk we will present several recent results on criteria ensuring the confinement of a quantum or a stochastic particle to a bounded domain in [math]\mathbb{R}^n[/math]. These criteria are given in terms of explicit growth and/or decay rates for the diffusion matrix and the drift potential close to the boundary of the domain. As an application of the general method, we will discuss several cases, including some where the background Riemannian manifold (induced by the diffusion matrix) is geodesically incomplete. These results are part of an ongoing joint project with G. Nenciu (IMAR, Bucharest, Romania).

## Friday, February 5, Daniele Cappelletti, Copenhagen University, speaks in the Applied Math Seminar, 2:25pm in Room 901

**Note:** Daniele Cappelletti is speaking in the Applied Math Seminar, but his research on stochastic reaction networks uses probability theory and is related to work of our own David Anderson.

Title: **Deterministic and Stochastic Reaction Networks**

Abstract: Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.

## Thursday, February 25, Ramon van Handel, ORFE and PACM, Princeton

## Thursday, March 3, Chris Janjigian, UW-Madison

Title: **Large deviations for certain inhomogeneous corner growth models**

Abstract: The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.

This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)