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Sensitivity of mixing times of Cayley graphs

2023; Cambridge University Press; Volume: 76; Issue: 4 Linguagem: Inglês

10.4153/s0008414x23000421

1496-4279

Jonathan Hermon, Gady Kozma,

Graph theory and applications

Abstract We show that the total variation mixing time is not quasi-isometry invariant, even for Cayley graphs. Namely, we construct a sequence of pairs of Cayley graphs with maps between them that twist the metric in a bounded way, while the ratio of the two mixing times goes to infinity. The Cayley graphs serving as an example have unbounded degrees. For non-transitive graphs, we construct bounded degree graphs for which the mixing time from the worst starting point for one graph is asymptotically smaller than the mixing time from the best starting point of the random walk on a network obtained by increasing some of the edge weights from 1 to $1+o(1)$ .