PROPORTION OF UNAFFECTED SITES IN A REACTION-DIFFUSION PROCESS

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 --> 0 when they meet. We find that, asymptotically, P(t) similar to t(-theta) with a universal exponent theta = 1/2 - 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 --> 0 with k > 2. In this case, a stretched exponential behaviour is found for all d greater than or equal to 1, except in the case k = 3, d = 1, where P(t) similar to e(-const(ln t)3/2.