4:00 pm Monday, November 24, 2014
Colloquium: Random walks on groups and the Kaimanovich-Vershik conjecture by
Russell Lyons (Indiana University) in RLM 5.104
Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if y is in xS. Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derriennic and Avez, showed that G admits non-constant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik also presented the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; in dimensions 5 and above, their conjecture was proved by Erschler (2011). I will discuss the background and present a simple proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Yuval Peres. Submitted by
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