  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 $n$.
