# Structure of the particle population for a branching random walk with a critical reproduction law

## TL;DR

This paper investigates the structure of particle populations in a symmetric branching random walk on a lattice, revealing how subpopulation behaviors influence convergence and clustering in low dimensions.

## Contribution

It provides new insights into the subpopulation structure and explains the effects of vanishing subpopulations on steady state convergence and cluster formation.

## Key findings

- Vanishing subpopulations do not hinder convergence to steady state.
- Clusterization occurs in dimensions 1 and 2 due to subpopulation dynamics.
- The structure of subpopulations explains the behavior of the entire process.

## Abstract

We consider a continuous-time symmetric branching random walk on the $d$-dimensional lattice, $d\ge 1$, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a critical Bienamye-Galton-Watson process at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point $x$. We answer why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions $d=1$ and $d=2$.