Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition

TL;DR

This paper derives a formula for the distribution of a tagged particle in a symmetric exclusion process with two-sided Bernoulli initial conditions, using integrable probability techniques and Fredholm determinants.

Contribution

It provides a new exact formula for the tagged particle distribution and large deviation function in the symmetric simple exclusion process with general initial densities.

Findings

Derived a Fredholm determinant formula for the current's moment generating function.

Obtained the large deviation function for the tagged particle position.

Extended results to the stationary measure with uniform density.

Abstract

For the two-sided Bernoulli initial condition with density $\rho_-$ (resp. $\rho_+$) to the left (resp. to the right), we study the distribution of a tagged particle in the one dimensional symmetric simple exclusion process. We obtain a formula for the moment generating function of the associated current in terms of a Fredholm determinant. Our arguments are based on a combination of techniques from integrable probability which have been developed recently for studying the asymmetric exclusion process and a subsequent intricate symmetric limit. An expression for the large deviation function of the tagged particle position is obtained, including the case of the stationary measure with uniform density $\rho$.