Exact solution of interacting particle systems related to random matrices

TL;DR

This paper derives explicit formulas for the distributions and correlation functions of one-dimensional interacting diffusions, including Brownian TASEP and non-colliding diffusions, connecting particle systems to random matrix theory.

Contribution

It provides the first explicit Fredholm determinant formulas for space-inhomogeneous particle systems with arbitrary initial conditions.

Findings

Finite dimensional distributions expressed as Fredholm determinants.

Determinantal correlation functions for non-colliding diffusions.

Applicable to models related to classical random matrix ensembles.

Abstract

We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.