TY - JOUR

T1 - Internal DLA in a random environment

AU - Ben Arous, Gérard

AU - Quastel, Jeremy

AU - Ramírez, Alejandro F.

N1 - Funding Information:
∗Corresponding author. E-mail addresses: Gerard.Benarous@epfl.ch (G. Ben Arous), quastel@math.utoronto.edu (J. Quastel), aramirez@mat.puc.cl (A.F. Ramírez). 1Partially supported by Natural Sciences and Engineering Research Council of Canada, and Alfred P. Sloan Foundation. 2Partially supported by Fundación Andes, DIPUC and Fondo Nacional de Desarrollo Científico y Tecnológico, grant 1990437.

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

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U2 - 10.1016/S0246-0203(02)00003-1

DO - 10.1016/S0246-0203(02)00003-1

M3 - Article

AN - SCOPUS:0037333721

SN - 0246-0203

VL - 39

SP - 301

EP - 324

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 2

ER -