A Fredholm Determinant Representation in ASEP

TL;DR

This paper derives a Fredholm determinant representation for the distribution of the m'th particle in ASEP with step initial condition, simplifying the integral series and enabling analysis of scaling limits.

Contribution

It introduces a Fredholm determinant formula for ASEP particle distributions, simplifying previous integral series and facilitating limit analysis.

Findings

Series summed to a single integral with Fredholm determinant

Derived a non-rigorous scaling limit using the determinant representation

Provided a new analytical tool for studying ASEP dynamics

Abstract

In previous work the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers, and consider the distribution function for the m'th particle from the left. In the previous work an infinite series of multiple integrals was derived for this distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.