  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 $d=3$.
