Proportion of Unaffected Sites in a Reaction-Diffusion Process

TL;DR

This paper analyzes the probability that a site remains unvisited in a reaction-diffusion process, revealing universal and non-universal decay exponents across different dimensions and reactions using field-theoretic methods.

Contribution

It provides a detailed field-theoretic analysis of site unvisited probabilities in reaction-diffusion systems, including universal exponents and behavior for multiple reaction types.

Findings

Universal decay exponent in 2D near criticality

Non-universal exponents depend on reaction rate in higher dimensions

Stretched exponential decay in general reaction-diffusion processes

Abstract

We consider the probability $P(t)$ that a given site remains unvisited by any of a set of random walkers in $d$ dimensions undergoing the reaction $A+A\to0$ when they meet. We find that asymptotically $P(t)\sim t^{-\theta}$ with a universal exponent $\theta=\ffrac12-O(\epsilon)$ for $d=2-\epsilon$, while, for $d>2$, $\theta$ is non-universal and depends on the reaction rate. The analysis, which uses field-theoretic renormalisation group methods, is also applied to the reaction $kA\to0$ with $k>2$. In this case, a stretched exponential behaviour is found for all $d\geq1$, except in the case $k=3$, $d=1$, where $P(t)\sim {\rm e}^{-\const (\ln t)^{3/2}}$.