Fluctuations of particle systems determined by Schur generating functions

TL;DR

This paper introduces a new analytical framework using Schur generating functions to study the global fluctuations of stochastic particle systems, establishing CLTs for various models in physics and representation theory.

Contribution

It develops a novel toolbox based on Schur generating functions for analyzing the asymptotic behavior of discrete particle systems, including proving CLTs for multiple models.

Findings

Established CLTs for random lozenge and domino tilings

Proved CLTs for non-intersecting random walks

Analyzed tensor product decompositions of unitary group representations

Abstract

We develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems. We introduce and study the notion of the Schur generating function of a random discrete configuration. Our main result provides a Central Limit Theorem (CLT) for such a configuration given certain conditions on the Schur generating function. As applications of this approach, we prove CLT's for several probabilistic models coming from asymptotic representation theory and statistical physics, including random lozenge and domino tilings, non-intersecting random walks, decompositions of tensor products of representations of unitary groups.