AI 中文总结
本文确定了奇阶阿贝尔群上随机Cayley图的直径临界窗口，证明覆盖时间收敛到泊松过程，直径转变呈Gumbel分布，并给出精确阈值常数\\(d!/2^d\\)及\\(d=2\\)时的路径-环分解。
AI 中文摘要
设\\(d\ge2\\)固定，并设\\(G_n\\)为阶数\\(N_n\to\infty\\)的有限奇阶阿贝尔群。我们确定了标准随机Cayley图中以直径为中心的\\(d\\)-临界窗口，其中每个非零群元素被独立选取。记\\(M_n=(N_n-1)/2\\)，我们证明归一化的首次\\(d\\)-距离覆盖时间满足\\[\sum_{[x]\in(G_n\setminus\{0\})/\{\pm1\}} \delta_{\frac{N_n^{d-1}}{d!}\tau_{n,[x]}^d-\log M_n} \xrightarrow{d} \PPP(e^{-z} \\,dz).\\]因此，临界窗口中对跖缺陷的数量依全变差收敛到泊松分布，直径转变具有Gumbel分布轮廓\\(e^{-e^{-c}}\\)，且直径命中时间具有Gumbel波动。在原始生成元密度参数化下，这在整个奇阶阿贝尔类中给出了精确的固定\\(d\\)阈值常数\\(d!/2^d\\)。对于\\(d=2\\)，我们还获得了目标表示图的精确路径-环分解。
英文摘要
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.
Comments0
