Front propagation in an exclusion one-dimensional reactive dynamics

TL;DR

This paper studies a one-dimensional exclusion process modeling reactive front propagation, proving laws of large numbers, central limit theorems, and convergence to an invariant measure for the front and particle distribution.

Contribution

It introduces a new analysis of reactive exclusion dynamics, establishing rigorous probabilistic results and a novel approach to defining regeneration times.

Findings

Law of large numbers for the front position

Central limit theorem for front fluctuations

Convergence to a unique invariant measure

Abstract

We consider an exclusion process representing a reactive dynamics of a pulled front on the integer lattice, describing the dynamics of first class $X$ particles moving as a simple symmetric exclusion process, and static second class $Y$ particles. When an $X$ particle jumps to a site with a $Y$ particle, their position is intechanged and the $Y$ particle becomes an $X$ one. Initially, there is an arbitrary configuration of $X$ particles at sites $..., -1,0$, and $Y$ particles only at sites $1,2,...$, with a product Bernoulli law of parameter $\rho,0<\rho<1$. We prove a law of large numbers and a central limit theorem for the front defined by the right-most visited site of the $X$ particles at time $t$. These results corroborate Monte-Carlo simulations performed in a similar context. We also prove that the law of the $X$ particles as seen from the front converges to a unique invariant measure. The proofs use regeneration times: we present a direct way to define them within this context.