Let Gamma be a countable group. We can try to measure the dimension of a subset of Gamma using the limiting exponent of the hitting probability for a fixed random walk on Gamma. As is shown in Kesten's PhD thesis, this works and is quite meaningful for subgroups. We prove that the limiting exponent also exists for invariant random partitions of Gamma, a natural generalization of invariant random subgroups that also includes iid percolation clusters and much more. The existence is surprisingly hard to prove as it does not seem to follow from the usual pointwise ergodic theorems. The result is already new for iid percolation clusters.

As a byproduct, we show that the growth of a unimodular random rooted tree of bounded degree always exists, assuming its upper growth passes the critical threshold of amenability. This complements Timar's work who showed the possible nonexistence of growth below this threshold. We also show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree.

This is joint work with Mikolaj Fraczyk and Ben Hayes.