2019/11/07 by Jonathan Hermon, Hermon, Jonathan, Sam Olesker-Taylor +1
Mathematics · #05C12 #05C80 #05C81 #20D15 #60J27 #60K37 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Limits and Structures in Graph Theory #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1911.02975
openalex publication_date 2019/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the random Cayley graph of a finite group G with respect to k generators chosen uniformly at random, with 1 ≪ k \lesssim log |G|. The results of this article supplement those in the three main papers on random Cayley graphs. The majority of the results are inspired by a `universality' conjecture of Aldous and Diaconis (1985). To start, we study the limit profile of cutoff for the simple random walk on this random graph, as well as a detailed investigation into mixing properties when G = \mathbb Zpd with p prime. We then exposit a proof of Diaconis and Saloff-Coste (1994) establishing lack of cutoff when k \asymp 1. We move onto discussing material from our companion paper on matrix groups. We then study distance of a typical element of G from the identity in an Lq-type graph distance in the Abelian set-up. Finally, we give necessary and sufficient conditions for k independent uniform elements of G to generate G, ie for the random Cayley graph to be connected, based on work of Pomerance (2001). The aforementioned results all hold with high probability over the random Cayley graph.