Statistical properties of the distance between a trapping center and a uniform density of diffusing particles in two dimensions

Several analyses of self-segregation properties of reaction-diffusion systems in low dimensions have been based on a simplified model in which an initially uniform concentration of point particles is depleted by reaction with an immobilized trap. A measure of self-segregation in this system is the distance of the trap from the nearest untrapped particle. In one dimension the average of this distance has been shown to increase at a rate proportional to t ^{1 4}. We show that this rate in a two-dimensional system is asymptotically proportional to (In t) ^{1 2}, and that the concentration profile in the neighborhood of the trap is proportional to (ln r ln t).