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Internal DLA in a random environment. (English) Zbl 1023.60089
In internal DLA (diffusion limited aggregation) traps are randomly distributed on \(\mathbb{Z}^d\) \((d\geq 1)\) and particles are injected at the origin, which perform independent random walks, until they hit an unsaturated trap, which then becomes saturated and the particle dies. The authors study the asymptotic shape and growth rate of the random cluster of saturated traps. They show that the large scale effect of the traps on the growth rate depends on the strength of the injection and identify different regimes. When the injection is subcritical, the set of saturated traps is asymptotically a ball and essentially all particles have been trapped. Thus the radius of the ball can be determined easily from the density of traps. This case was proved by two of the authors [G. Ben Arous and A. F. Ramírez, Ann. Probab. 28, 1470-1527 (2000)].
In this article the critical and supercritical cases are treated. In the critical regime the shape is still a ball, but with a nonvanishing density of live particles inside the ball and a nontrivial dependence of the growth rate on the density of traps. This case is proved by modifying the hydrodynamic limit method of J. Gravner and J. Quastel [ibid. 28, 1528-1562 (2000)]. The supercritical regime is studied using order statistics, which is restricted to \(d=1\). In the supercritical, subexponential regime the traps slow down the growth rate, but in a way which does not depend on their density. In the supercritical, superexponential regime there is no effect of the traps at all.

MSC:
60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60G70 Extreme value theory; extremal stochastic processes
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