Shape and local growth for multidimensional branching random walks in random environment

TL;DR

This paper investigates the behavior of multidimensional branching random walks in random environments, establishing limit theorems for visited sites, local population size, and total particle count, revealing a compact shape and growth dynamics.

Contribution

It introduces new limit theorems for the shape and local growth of branching random walks in random environments with finite range dependence.

Findings

Limiting shape of visited sites is compact and convex

Local population size exhibits concave growth

Law of large numbers for total particle count

Abstract

We study branching random walks in random environment on the $d$-dimensional square lattice, $d \geq 1$. In this model, the environment has finite 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, though 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.