On explosions in heavy-tailed branching random walks

TL;DR

This paper characterizes when explosion occurs in heavy-tailed branching random walks, linking it to the finiteness of summed minimal displacements, and explores particle positions when explosion does not happen.

Contribution

It provides a precise characterization of explosion conditions in heavy-tailed branching random walks and establishes the optimality of the tail condition for this equivalence.

Findings

Explosion occurs if and only if the sum of minimal displacements over generations is finite.

The tail condition for explosion is shown to be optimal.

When no explosion, the minimal particle position diverges to infinity.

Abstract

Consider a branching random walk on $\mathbb{R}$, with offspring distribution Z and nonnegative displacement distribution W. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution Z is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs (Z, W) for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_n$, the position of the particle in generation n closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_n=\infty$).