Mixing in groups: From Ramanujan graphs to the product replacement algorithm

Abstract

I will survey several recent works involving mixing for random walks on groups and regular graphs. In joint work with Eyal Lubetzky, we determined precisely the mixing time of random walk on optimal d-regular expanders (Ramanujan graphs) and showed these walks exhibit cutoff. Much less is known for the random walk on n by n invertible matrices mod 2 obtained by adding a random row to another random row; the spectral gap was found by Kassabov (2005) but the mixing time is unknown. The computational mixing time in this group was related by Rivest and Sotiraki to a cryptographic signature scheme. If we focus on one column in these matrices, we obtain a random walk on the hypercube considered by Chung and Graham (1997), who gave an upper bound of O(n\log n) for the mixing time; this was refined to cutoff at time (3/2) n\log n+O(n) in joint work with Anna Ben Hamou (2017). Recently, (jointly with R. Tanaka and A. Zhai) we extended the latter result to any fixed set of columns, and more generally, to the product replacement algorithm on any finite group; surprisingly, the leading constant 3/2 persists.