Internal DLA in a random environment

In this article, Internal DLA is studied with a random, homogeneous, distribution of traps. Particles are injected at the origin of a d-dimensional Euclidean lattice and perform independent random walks until they hit an unsaturated trap, at which time the particle dies and the trap becomes saturated. It is proved that the large scale effect of the randomness of the traps on the speed of growth of the set of saturated traps depends of the strength of the injection, and separates into several regimes. In the subcritical regime, the set of saturated traps is asymptotically an Euclidean ball whose radius is determined in a trivial way from the trap density. In the critical regime, there is a nontrivial interplay between the density of traps and the rate of growth of the ball. The supercritical regime is studied using order statistics for free random walks. This restricts us to d = 1. In the supercritical, subexponential regime, there is an overall effect of the traps, but their density does not affect the growth rate. Finally, in the supercritical, superexponential regime, the traps have no effect at all, and the asymptotics is governed by that of free random walks on the lattice. 2003 Éditions scientifiques et médicales Elsevier SAS.