For random d-regular graphs on N vertices with \(1 \ll d \ll N^{2/3}\), we develop a \(d^{-1/2}\) expansion of the local eigenvalue distribution about the Kesten–McKay law up to order \(d^{-3}\). This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erdős–Rényi graphs of the same average degree. As a first application, for \(1 \ll d \ll N^{2/3}\), we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by \((2 + {{\,\mathrm{o}\,}}(1)) \sqrt{d - 1}\). As a second application, for \(N^{2/9} \ll d \ll N^{1/3}\), we prove that the extremal eigenvalues are concentrated at scale \(N^{-2/3}\) and their fluctuations are governed by Tracy–Widom statistics. Thus, in the same regime of d, \(52\%\) of all d-regular graphs have second-largest eigenvalue strictly less than \(2 \sqrt{d - 1}\).

中文翻译：

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中度随机正则图的边刚度和通用性

对于具有\（1 \ ll d \ ll N ^ {2/3} \）的N个顶点上的随机d-正则图，我们开发了局部特征值分布的\（d ^ {-1/2} \）展开约Kesten–McKay定律直到\（d ^ {-3} \）为止。该结果在频谱边缘一直有效。这暗示着，这些随机正则图的特征值比相同平均度数的Erdős-Rényi图的特征值更严格。作为第一个应用，对于\（1 \ ll d \ ll N ^ {2/3} \），我们证明了邻接矩阵的所有非平凡特征值都具有很高的概率，其绝对值以\（（2 + { {\，\ mathrm {o} \，}}（1））\ sqrt {d-1} \）。作为第二个应用\（N ^ {2/9} \ ll d \ ll N ^ {1/3} \），我们证明极值特征值集中在标度\（N ^ {-2/3} \）上，并且它们的波动是由Tracy-Widom统计数据管理。因此，在相同的d范围内，所有d-正则图的\（52 \％\）的第二大特征值严格小于\（2 \ sqrt {d-1} \）。