Analysis and Probability seminar

Abstract: The stochastic six-vertex model is a discrete time interacting particle system obtained from the six-vertex model by restricting its parameters. In this model particles have a tendency to move to the right. Starting from a blocked configuration, i.e. one where all positions to the right of the origin are occupied, and all to the left of the origin are empty, the dynamics is recurrent and the stationary measure is given by a so called "blocking measure".

In an upcoming work, we show that the convergence of the process to this stationary measure is one-sided, in the sense that at every intermediate time the law of the process is stochastically dominated by the stationary law, in the ordering of particle configurations given by height functions. We further show that this even holds when the dynamics of the particle system is modified by disallowing certain transitions at certain times. Statements like this have been useful in the study of the behaviour of second class particles and of mixing times for other models.

Analogous domination results are known for particle systems with a certain "monotonicity" property, however the stochastic six-vertex process lacks this property. We instead use a novel connection of the stochastic six-vertex model to parabolic Kazhdan-Lusztig R polynomials, which have are polynomials indexed by two partitions which have previously appeared in representation theory.

In the talk I will introduce the model, explain the statement and give a combinatorial description of the parabolic Kazhdan-Lusztig R polynomials. If time permits I will then give the connection of the stochastic six-vertex model to Iwatori-Hecke algebras which explains the appearance of these polynomials.