Heat kernels are not uniform expanders

We study infinite analogues of expander graphs, namely graphs whose subgraphs weighted by heat kernels form an expander family. Our main result is that there does not exist any infinite expander in this sense. This proves the analogue for random walks of Benjamini’s conjecture that there is no infinite graph whose metric balls are uniformly expanders. The proof relies on a study of stationary random graphs, in particular proving non-expansion of heat kernels in that setting. A key result is that any stationary random graph is stationary hyperfinite, which is a new property of independent interest.