Nearest-neighbor distance at a single mobile trap

We consider a one-dimensional system with an infinite number of identical particles A diffusing in the presence of a single diffusing perfect trap [Formula Presented] We study numerically the average distance [Formula Presented] from the trap to the nearest unreacted [Formula Presented] and confirm the claim that in the long-time limit [Formula Presented] where [Formula Presented] is an exponent depending on the ratio of diffusivities [Formula Presented] and [Formula Presented] of the particles A and the trap [Formula Presented] respectively. We also confirm the validity of a conjecture for the value of [Formula Presented] but show that it should be limited to a specific choice of the initial distribution of particles [Formula Presented]