Consider the random Cayley graph of a finite group G with respect to k generators chosen uniformly at random, with 1 << log k << log|G|: the vertices are the group elements, and g, h in G are connected if there exists a generator z so that g = hz or gz = h.
A conjecture of Aldous and Diaconis asserts that the simple random walk on this graph exhibits cutoff, at a time which depends only on |G| and k, not on the algebraic structure of the group G (ie universally in G). We verify this conjecture for a wide class of Abelian groups.
Time permitting, we discuss extensions to the case where the underlying group G is non-Abelian. There the cutoff time cannot be written only as a function of |G| and of k; it depends on the algebraic structure.
Joint work with Jonathan Hermon