On ASEP with Periodic Step Bernoulli Initial Condition

TL;DR

This paper studies ASEP with a periodic initial density pattern, deriving a formula for the probability distribution of particles over time, extending previous results from step Bernoulli to periodic cases.

Contribution

It generalizes the probability distribution formula for ASEP to initial conditions with periodic density functions, expanding understanding beyond constant step Bernoulli cases.

Findings

Derived a sum-over-integers formula for particle probabilities

Extended previous step Bernoulli results to periodic initial conditions

Provided integral representations for probability distributions

Abstract

We consider the asymmetric simple exclusion process (ASEP) on the integers in which the initial density at a site (the probability that it is occupied) is given by a periodic function on the positive integers. (When the function is constant this is the step Bernoulli initial condition.) Starting with a result in earlier work we find a formula for the probability distribution for a given particle at a given time which is a sum over positive integers k of integrals of order k.