Fredholm determinants in the multiparticle hopping asymmetric diffusion model

TL;DR

This paper derives Fredholm determinant formulas for the probability distributions of particle positions in the multiparticle hopping asymmetric diffusion model and the PushASEP, providing exact solutions for these stochastic processes.

Contribution

It introduces Fredholm determinant representations for the position distributions in MADM and PushASEP with specific initial conditions, advancing exact analytical understanding.

Findings

Fredholm determinant representation for MADM particle positions

Fredholm determinant for PushASEP with step Bernoulli initial condition

Exact formulas for particle position probabilities

Abstract

In this paper we treat the multiparticle hopping asymmetric diffusion model (MADM) of which initial configuration is such that a single site is occupied by infinitely many particles and all other sites are empty. We show that the probability distribution of the $m^{\textrm{th}}$ leftmost particle's position at time $t$ is represented by a Fredholm determinant. Also, we consider an exclusion process type model of the MADM, which is the (two-sided) PushASEP. For the PushASEP with the step Bernoulli initial condition, we find a Fredholm determinant representation of the probability distribution of the $m^{\textrm{th}}$ leftmost particle's position at $t$.