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We study infinite analogues of expander graphs, namely graphs where subgraphsweighted by heat kernels form an expander family. Our main result is that theredoes not exist any infinite expander in this sense. This proves the analoguefor random walks of Benjamini's conjecture that there is no infinite graphwhose metric balls are uniformly expander. The proof relies on a study ofstationary random graphs, in particular proving non-expansion of heat kernelsin that setting. A key result is that any stationary random graph is stationaryhyperfinite, which is potentially of independent interest.

Heat kernels are not uniform expanders