Branching random walks conditioned on rarely survival

TL;DR

This paper investigates the behavior of branching random walks conditioned on rare survival events, analyzing the convergence of Galton-Watson trees and studying their span and gap statistics under various offspring and displacement distributions.

Contribution

It introduces a convergence result for conditioned Galton-Watson trees and extends the analysis of span and gap statistics to more general branching random walks.

Findings

Galton-Watson trees conditioned on fixed particle number converge as size grows

Span and gap statistics are characterized for generalized branching random walks

Results extend previous models to arbitrary offspring and displacement distributions

Abstract

In this paper, we show that a Galton-Watson tree conditioned to have a fixed number of particles in generation $n$ converges in distribution as $n\rightarrow\infty$, and with this tool we study the span and gap statistics of a branching random walk on such trees, which is the discrete version of Ramola, Majumdar and Schehr 2015, generalized to arbitrary offspring and displacement distributions with moment constraints.