Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups

Abstract

Let \(d\ge2\) be fixed and let \(G_n\) be finite abelian groups of odd orders \(N_n\to\infty\). We determine the centered diameter-\(d\) critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing \(M_n=(N_n-1)/2\), we prove that the normalized first distance-\(d\) coverage times satisfy \sum_{[x]\in(G_n\setminus\{0\})/\{\pm1\}} δ_{\frac{N_n^{d-1}}{d!}τ_{n,[x]}^d-\log M_n} \xrightarrow{d} \PPP(e^{-z} $\,dz). Consequently, the number of antipodal defects in the critical window converges in total variation to a Poisson law, the diameter transition has the Gumbel profile \(e^{-e^{-c}}\), and the diameter hitting time has Gumbel fluctuations. In the original generator-density parametrization this yields the sharp fixed-\(d\) threshold constant \(d!/2^d\) throughout the odd-order abelian class. For \(d=2\), we additionally obtain an exact path--cycle decomposition of the target representation graphs.