Proportion of unaffected sites in a reaction-diffusion process

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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) approximately t- phi with a universal exponent theta = 1/2 approximately 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 renormalization-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>or=1, except in the case k=3, d=1, where P(t) approximately e(-const(Int)32/).