On the trapping reaction with mobile traps

We present Monte Carlo results for the two-species trapping reaction $A+B \to B$ with diffusing $A$ and $B$ on lattices in one, two and three dimension. We use a novel algorithm which permits to simulate survival probabilities of $A$ partic les down to $<10^{-30}$ with high accuracy. The results for the survival probability agree much better with the exact asymptotic predictions of Bramson and Lebowitz (Phys. Rev. Lett. {\bf 61}, 2397 (1988)) than with the heuristics of Kang and Redner (J. Phys. A {\bf 17}, L451 (1984)). But there are very large deviations from either which show that even these simulations are far from asymptotia. This is supported by the rms. displacement of $A$ particles which clearly show that the asymptotic regime has not been reached, at least for $d=2$ and $d=3$.