Limiting one-point distribution of periodic TASEP

TL;DR

This paper investigates the limiting one-point distribution of the periodic TASEP, revealing its dependence on scaled time, its transition from Tracy-Widom to Gaussian distribution, and its connection to integrable differential equations.

Contribution

It characterizes the time-dependent limiting distribution of periodic TASEP and links it to well-known integrable systems, extending understanding beyond the infinite line case.

Findings

Distribution transitions from Tracy-Widom to Gaussian over time

Established tail estimates for the distribution at all times

Connected the distribution to integrable differential equations such as KP, mKdV, and KdV

Abstract

The relaxation time limit of the one-point distribution of the spatially periodic totally asymmetric simple exclusion process is expected to be the universal one point distribution for the models in the KPZ universality class in a periodic domain. Unlike the infinite line case, the limiting one point distribution depends non-trivially on the scaled time parameter. We study several properties of this distribution for the case of the periodic step and flat initial conditions. We show that the distribution changes from a Tracy-Widom distribution in the small time limit to the Gaussian distribution in the large time limit, and also obtain right tail estimate for all time. Furthermore, we establish a connection to integrable differential equations such as the KP equation, coupled systems of mKdV and nonlinear heat equations, and the KdV equation.