Intermittency for branching walks with heavy tails

TL;DR

This paper studies branching random walks with heavy-tailed jumps, revealing that the front propagates exponentially fast and particles exhibit intermittent, highly non-uniform distribution behind the front, with a quantifiable zone of uniformity.

Contribution

It introduces a detailed analysis of intermittency and front propagation in heavy-tailed branching walks, including the rate at which the uniform zone expands.

Findings

Front propagates exponentially fast.

Particles are concentrated in sparse spots behind the front.

The non-uniform zone expands at a power rate.

Abstract

Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e., the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.