Heat kernels are not uniform expanders

Mikolaj Fraczyk; Wouter van Limbeek

Heat kernels are not uniform expanders

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arxiv 1905.13584 v1 pith:J4QJT3XY submitted 2019-05-31 math.CO math.DSmath.MG

Mikolaj Fraczyk , Wouter van Limbeek This is my paper · ORCID

classification math.COmath.DSmath.MG

keywords expandergraphsheatinfinitekernelsrandomstationarygraph

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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.

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Exponentially slow thermalization in 1D fragmented dynamics
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Exponential fragmentation of Hilbert space in 1D constrained dynamics implies exponentially slow thermalization under a boundary bath, with proofs for several model classes and a reduction to Benjamini's expander conjecture.

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