  We study infinite analogues of expander graphs, namely graphs where 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 expander. 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 potentially of independent interest.
