We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes G = G(n), whose ranks and nilpotency classes are uniformly bounded.  For some k = k(n) such that 1 << log k << log |G|, we pick a random set of generators S = S(n) by sampling k elements Z_1, …, Z_k from G uniformly at random with replacement, and set S = { Z_j^{\pm 1}: 1 <= j <= k }.  We show that the simple random walk on Cay(G,S) exhibits cutoff with high probability.  Some of our results apply to a general set of generators.  Namely, we show that there is a constant c > 0, depending only on the rank and the nilpotency class of G, such that for all symmetric sets of generators S of size at most c log |G| / log log |G|, the spectral gap and the mixing time of the simple random walk X = (X_t) on Cay(G,S) are asymptotically the same as those of the projection of X to the abelianization of G, given by [G,G]X_t.  In particular, X exhibits cutoff if and only if its projection does.  Based on joint work with Jonathan Hermon.

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