# Fluctuations of propagation front in catalytic branching walk

## TL;DR

This paper investigates the behavior and fluctuations of the propagation front in supercritical catalytic branching random walks on multidimensional lattices, extending previous one-dimensional results to higher dimensions and multiple catalysts.

## Contribution

It generalizes fluctuation analysis of the propagation front from one-dimensional to multidimensional CBRW and handles cases with multiple catalysts, using new probabilistic-analytic methods.

## Key findings

- Established fluctuation results for the propagation front in multidimensional CBRW.
- Extended Carmona and Hu's one-dimensional results to higher dimensions.
- Analyzed the case of multiple catalysts in one-dimensional CBRW.

## Abstract

We consider a supercritical catalytic branching random walk (CBRW) on a multidimensional lattice Z^d (d is positive integer). The main subject of study is the behavior of particles cloud in space and time. For CBRW on an integer line, Carmona and Hu (2014) examined the asymptotical behavior of the maximal coordinate M_n of the particles at time n. They proved that M_n/n converges to \mu almost surely (on a set of local non-degeneracy of CBRW), as n tends to infinity, where \mu>0 is a certain constant. Under additional assumption of a single catalyst in CBRW they also investigated the fluctuations of M_n with respect to \mu n, as n grows to infinity. Bulinskaya (2018) extended the strong limit theorem by Carmona and Hu having estimated the rate of the population propagation for the front of a multidimensional CBRW. Now our aim is to analyze fluctuations of the propagation front in CBRW on Z^d. We not only solve the problem in a multidimensional setting but also, treating the case of an arbitrary finite number of catalysts for d = 1, generalize the result by Carmona and Hu with the help of other probabilistic-analytic methods. Keywords and phrases: catalytic branching random walk, supercritical regime, spread of population, propagation front, fluctuations of front.