Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive

TL;DR

This paper studies the behavior of simple random walks on Galton-Watson trees with offspring distributions in the domain of attraction of stable laws, revealing spectral dimensions and fluctuation behaviors depending on the stability index.

Contribution

It provides new results on spectral dimensions and fluctuation phenomena for random walks on conditioned Galton-Watson trees with infinite variance offspring distributions.

Findings

Spectral dimension is 2α/(2α-1) for offspring distribution index α

Logarithmic fluctuations in quenched transition density for α in (1,2)

Tail bounds for generation size and total individuals in conditioned Galton-Watson trees

Abstract

We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, $Z$ say, is in the domain of attraction of a stable law with index $\alpha\in(1,2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $2\alpha/(2\alpha-1)$. Furthermore, we demonstrate that when $\alpha\in(1,2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $\alpha=2$. In the course of our arguments, we obtain tail bounds for the distribution of the $n$th generation size of a Galton-Watson branching process with offspring distribution $Z$ conditioned to survive, as well as tail bounds for the distribution of the total number of individuals born up to the $n$th generation, that are uniform in $n$.