Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

TL;DR

This paper analyzes the large-time fluctuations of particle crossings in the stationary TASEP, revealing a new family of distribution functions related to the GUE Tracy-Widom distribution through Fredholm determinants.

Contribution

It establishes the asymptotic distribution of space-time covariance fluctuations in stationary TASEP using Fredholm determinant asymptotics, extending previous results from the PNG model.

Findings

Fluctuations are of order t^{1/3} for large t.

The limiting distribution is a family F_w(s), a generalization of Tracy-Widom.

F_w(s) relates to the space-time covariance and second class particle probabilities.

Abstract

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \rho measure as initial conditions, 0<\rho<1, is stationary in space and time. Let N_t(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For j=(1-2\rho)t+2w(\rho(1-\rho))^{1/3} t^{2/3} we prove that the fluctuations of N_t(j) for large t are of order t^{1/3} and we determine the limiting distribution function F_w(s), which is a generalization of the GUE Tracy-Widom distribution. The family F_w(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at F_w(s) through the asymptotics of a Fredholm determinant. F_w(s) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.