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

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 $α\in(1,2]$. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is $2α/(2α-1)$. Furthermore, we demonstrate that when $α\in(1,2)$ there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when $α=2$. 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 $n$.

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