Distribution of a particle's position in the ASEP with the {alternating} initial condition

TL;DR

This paper derives the distribution of a particle's position in the ASEP with alternating initial conditions, revealing new combinatorial identities and expressing the distribution in a determinantal form.

Contribution

It provides the first explicit distribution formula for ASEP with alternating initial conditions, including novel combinatorial identities linking integrands to determinantal forms.

Findings

Derived the distribution of particle positions in ASEP with alternating initial condition.

Identified new combinatorial identities related to the initial condition.

Expressed the distribution in a determinantal form with an additional product.

Abstract

In this paper we give the distribution of the position of the particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find $\mathbb{P}(X_m(t) \leq x)$ where $X_m(t)$ is the position of the particle at time $t$ which was at $m =2k-1, k \in \mathbb{Z}$ at $t=0.$ As in the ASEP with the step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand to a determinantal form together with an extra product.