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 $A+A\to 0$ when they meet. We find that asymptotically $P(t)\sim t^{-\theta }$ with a universal exponent $θ=\ffrac12-O(ε)$ 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 $kA\to 0$ with $k>2$. In this case, a stretched exponential behaviour is found for all $d\ge 1$, except in the case $k=3$, $d=1$, where $P(t)\sim {\rm e}^{-\const (\ln t)^{3/2}}$.

10 pages, (revised version with abstract included) OUTP-94-35S

References in corpus (1)

• Renormalization Group Calculation for the Reaction $kA\to \varnothing$

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