Heat kernels are not uniform expanders

Understand

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.