### Lower Bounds for the Escape Rate of Random Walks on Amenable Graphs

Probability Seminar

3rd May 2024, 3:30 pm – 4:00 pm

Fry Building, 2.04

This talk will be based on a 2013 paper of Lee and Peres, in which they give lower bounds for the escape rate of the simple symmetric random walk (SSRW) on vertex-transitive amenable graphs. The escape rate of a random walk is the rate at which the expected displacement of the walk increases over time.

It is known, due to Erschler, that for the SSRW on an amenable group, the escape rate is \Theta(t^d), for some d \geq 1/2. In the paper, Lee and Peres use existence of non-constant equivariant harmonic maps to prove that the escape rate is \Omega(t^d) for d = 1/2 on amenable graphs. I will begin by defining amenability, equivariance and harmonicity, before giving a brief overview of the proof, as well as discussing the relationship between the results for groups and graphs.

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