We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.
Since our interest is in the effect of heavy-tailed environments, we assume that the branching rates are bounded away from zero and thus avoid the issue of recurrence and transience. Indeed we see that as soon as a site is occupied, there are almost immediately exponentially many particles, and we focus on analysing the growth of the branching process by describing when sites are hit and how the number of particles evolves thereafter. We find that for our choice of potential the sites that are hit, as well as the local growth rates, are --- even after rescaling appropriately --- random. This is in sharp contrast to existing shape theorems for branching random walks with spatial i.i.d offspring distribution, see @cite_1 @cite_11 , where the set of visited sites converges after rescaling to a deterministic convex set and the local growth rates are given by a deterministic function. Furthermore, we will show that in our case the growth rates for the actual number of particles deviates dramatically from those for the expected number, which again contrasts with @cite_1 @cite_11 .
@cite_1 @cite_11
2000306115 2963863740
We study branching random walks in random i.i.d. environment in Z d , d ≥ 1. For this model, the population size cannot decrease, and a natural definition of recurrence is introduced. We prove a dichotomy for recurrence transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time. We study branching random walks in random environment on the d- dimensional square lattice, d 1. In this model, the environment has nite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, and the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process. 1. Introduction and results We start with an informal description of the model we study in this paper. Particles live in Z d and evolve in discrete time. At each time, every particle is substituted by (possibly more than one) ospring which are placed in neighboring sites, independently of the other particles. The rules of ospring generation depend only on the location of the particle. The collection of those rules (so-called the environment) is itself random, it is chosen randomly before starting the process, and then it is kept xed during all the subsequent evolution of the particle system.