# Catalytic Branching Random Walk with Semi-exponential Increments

## TL;DR

This paper investigates the spatial spread of particles in a catalytic branching random walk with semi-exponential jump distributions, revealing a faster growth rate and a non-convex limiting front shape in the supercritical regime.

## Contribution

It establishes a limit theorem for the normalized positions of particles at the front, highlighting differences from light-tailed cases and extending understanding to semi-exponential distributions.

## Key findings

- Normalizing factor grows faster than linear in time.
- Limiting front shape is non-convex.
- Power rate normalization contrasts with light tail cases.

## Abstract

A catalytic branching random walk on a multidimensional lattice, with arbitrary finite number of catalysts, is studied in supercritical regime. The dynamics of spatial spread of the particles population is examined, upon normalization. The components of the vector random walk jump are assumed independent (or close to independent) and have semi-exponential distributions with, possibly, different parameters. A limit theorem on the almost sure normalized positions of the particles at the population ``front'' is established. Contrary to the case of the random walk increments with ``light'' distribution tails, studied by Carmona and Hu (2014) in one-dimensional setting and Bulinskaya (2018) in multidimensional setting, the normalizing factor has a power rate and grows faster than linear in time function. The limiting shape of the front in the case of semi-exponential tails is non-convex in contrast to a convex one in the case of light tails.