The measurable Kesten theorem

2010. november 18. csütörtök 16:15

If the speed of the random walk on d-regular vertex transitive graph agrees with that on the d-regular tree, then the graph is itself a tree. This is a rigidity result, similar to Kesten's famous theorem, which states the same for spectral radius.

Unimodular networks are the random analogues of vertex-transitive graphs: they are random rooted graphs invariant under moving the root. In this case, the first rigidity result fails, while Kesten's theorem is still true.