Semi-infinite particle systems with exclusion interaction and heterogeneous jump rates

TL;DR

This paper analyzes semi-infinite particle systems on a 1D lattice with exclusion and heterogeneous jump rates, showing convergence to a stable distribution and a law of large numbers for particle positions.

Contribution

It establishes conditions under which particles form a stable cloud with a product-geometric distribution and a common speed, extending understanding of heterogeneous exclusion processes.

Findings

Particles form a stable, semi-infinite cloud.

Inter-particle distances converge to a product-geometric distribution.

Particles obey a strong law of large numbers with a shared speed.

Abstract

We study semi-infinite particle systems on the one-dimensional integer lattice, where each particle performs a continuous-time nearest-neighbour random walk, with jump rates intrinsic to each particle, subject to an exclusion interaction which suppresses jumps that would lead to more than one particle occupying any site. Under appropriate hypotheses on the jump rates (uniformly bounded rates is sufficient) and started from an initial condition that is a finite perturbation of the close-packed configuration, we give conditions under which the particles evolve as a single, semi-infinite "stable cloud". More precisely, we show that inter-particle separations converge to a product-geometric stationary distribution, and that the location of every particle obeys a strong law of large numbers with the same characteristic speed.