Multi-point distribution of discrete time periodic TASEP

TL;DR

This paper derives a multi-point distribution formula for discrete time TASEP on a periodic domain, analyzing large-time limits and boundary effects for various initial conditions, including step and flat.

Contribution

It provides a new multi-point joint distribution formula involving Fredholm determinants for discrete time TASEP with periodic boundary conditions, including large-time asymptotics.

Findings

Derived multi-point distribution formula involving Fredholm determinants.

Established large-time limit results for finite and infinite lattices.

Verified assumptions for step and flat initial conditions.

Abstract

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale $t=O(L^{3/2})$ when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice $\mathbb{Z}$ by taking the period $L$ large enough so that the finite-time distribution is not affected by the boundary. The large time limit for multi-time distribution for discrete time TASEP on $\mathbb{Z}$ is then obtained for the step initial condition.