Dynamics of the leftmost particle in heterogeneous semi-infinite exclusion systems

TL;DR

This paper investigates the long-term behavior of the leftmost particle in a semi-infinite exclusion process with heterogeneous jump rates, identifying conditions for recurrence or transience and analyzing the influence of initial conditions.

Contribution

It provides new criteria for recurrence and transience of the leftmost particle based on jump rates and initial conditions, including tools for analyzing its escape rate.

Findings

Leftmost particle can be null recurrent, positive recurrent, or transient.

Comparison with M/G/∞ queue helps analyze escape rates.

Initial conditions significantly influence the particle's dynamics.

Abstract

We study the behaviour of the leftmost particle in a semi-infinite particle system on $\mathbb{Z}$, where each particle performs a continuous-time nearest-neighbour random walk, with particle-specific jump rates, subject to the exclusion interaction (i.e., no more than one particle per site). We give conditions, in terms of the jump rates on the system, under which the leftmost particle is recurrent or transient, and develop tools to study its rate of escape in the transient case, including by comparison with an $M/G/\infty$ queue. In particular we show examples in which the leftmost particle can be null recurrent, positive recurrent, ballistically transient, or subdiffusively transient. Finally we indicate the role of the initial condition in determining the dynamics, and show, for example, that sub-ballistic transience can occur started from close-packed initial configurations but not from stationary initial conditions.