2024/03/19 by Jonathan Hermon, Hermon, Jonathan, Xiangying Huang +1 · 2 citations
Mathematics · #05C48 #05C80 #05C81 #20D15 #60B15 #60J27 #60K37 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2403.12355
openalex publication_date 2024/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Z1,…,Zk from G uniformly at random with replacement, and set S:=\Zj± 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 (clog |G|)/(log log |G|), the spectral gap and the ε-mixing time of the simple random walk X=(Xt)t≥ 0 on Cay(G,S) are asymptotically the same as those of the projection of X to the abelianization of G, given by [G,G]Xt. In particular, X exhibits cutoff if and only if its projection does.