Random Walks on Groups and the Kaimanovich-Vershik Conjecture for Lamplighter Groups

  • Yuval Peres | Microsoft Research Redmond

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 their ratio xy-1 is in S.
Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derrienic 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 (1983) also described 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 dimension 5 and above, their conjecture was proved by Anna Erschler (2011). In the talk, I will discuss the background and present a proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Russ Lyons; the case of dimension 3 is the most delicate. No prior knowledge of group theory will be assumed.

Speaker Details

Yuval Peres is a Principal Researcher in the Theory group at Microsoft Research, Redmond. His research encompasses many areas of probability theory including random walks, Brownian motion, percolation, point processes and random graphs, as well as connections with Ergodic Theory, PDE, Combinatorics, Fractals and Theoretical Computer Science; cf. http://arxiv.org/find/math/1/au:+Peres_Y/0/1/0/all/0/1

He has advised 20 PhD students, see http://www.genealogy.math.ndsu.nodak.edu/id.php?id=22523&fChrono=1

    • Portrait of Jeff Running

      Jeff Running

    • Portrait of Yuval Peres

      Yuval Peres

      Principal Researcher

Series: Microsoft Research Talks