Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups
Abstract
Let d≥2 be fixed and let Gn be finite abelian groups of odd orders Nn→∞. We determine the centered diameter-d critical window for the standard random Cayley graph in which each nonzero group element is selected independently. Writing Mn=(Nn−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!/2d throughout the odd-order abelian class. For d=2, we additionally obtain an exact path--cycle decomposition of the target representation graphs.
Related
- Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
- Fast robbers on abelian Cayley graphs and digraphs
- Isomorphisms of abelian Cayley graphs with their natural edge-colouring
- Optimal Sparsifiers for Abelian Cayley Graphs
- A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs
BibTeX
@misc{zhi2026exact,
title = {{Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups}},
author = {Yao Zhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28573},
eprint = {2609.28573},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28573},
}
RIS
TY - UNPB
TI - Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups
AU - Yao Zhi
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.28573
UR - https://arxiv.org/abs/2609.28573
AB - Let d≥2 be fixed and let G_n be finite abelian groups of odd orders N_n→∞. 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.
ER -