Proportion of Unaffected Sites in a Reaction-Diffusion Process

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 $P(t)\sim t^{-θ}$ with a universal exponent $θ=\ffrac12-O(ε)$ for $d=2-ε$, while, for $d>2$, $θ$ 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 $P(t)\sim {\rm e}^{-\const (\ln t)^{3/2}}$.