Advanced Topics in Probability Theory, WS 2019/20

Introduction to Integrable Probability

Time and Room

Wed 10-12, LWK 0.003 and Fri, 10-12, LWK 0.011

This is the current plan of the lecture

(1) Introduction. We will start by showing some of the models which can be threated by the method that we will learn during the lecture. These span from interacting particle systems, random tiling, stochastic growth models, equilibrium crystal shapes and random matrices.
(2) Schur processes. We will introduce a class of stochastic processes on partitions (which can be though also as evolution of some point processes).
(3) Correlation functions for Schur processes. We will see that Schur processes have a determinantal correlation structure.
(4) Applications. We will apply the results to some of the models mentioned in the Introduction, in particular the totally asymmetric simple exclusion process (TASEP).
(5) Duality method. For one or two generatization of TASEP, the interacting particle systems studied above, we will discuss the duality method, which allow to recover distribution functions.
(6) Limits and asymptotics. We will see how distribution functions of some observables of continuous models like the Stochastic Heat Equation (SHE) can be obtained the limit of the models in (5). Some hints on asymptotic results will be also presented.

The current lecture notes be found here. The last chapter is here.