Geometry of Random Cayley Graphs of Abelian Groups
Jonathan Hermon, Sam Olesker-Taylor

TL;DR
This paper studies the geometric and spectral properties of random Cayley graphs of finite Abelian groups, revealing concentration phenomena and asymptotic behaviors that depend only on the size of the group and the number of generators.
Contribution
It establishes concentration of distances, asymptotic diameter, and spectral gap estimates for random Abelian Cayley graphs, extending classical results to a broader setting.
Findings
Distance concentrates around a minimal radius M
Graph diameter asymptotically equals M in certain regimes
Spectral gap scales as |G|^{-2/k} under specific conditions
Abstract
Consider the random Cayley graph of a finite Abelian group with respect to generators chosen uniformly at random, with . Draw a vertex . We show that the graph distance from the identity to concentrates at a particular value , which is the minimal radius of a ball in of cardinality at least , under mild conditions. In other words, the distance from the identity for all but of the elements of lies in the interval . In the regime , we show that the diameter of the graph is also asymptotically . In the spirit of a conjecture of Aldous and Diaconis (1985), this depends only on and , not on the algebraic structure of . Write for the minimal size of a generating subset of . We…
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