标题： 固定度数的随机正则图的局部谱稳定性 显示英文标题

标题： Local spectral stability for random regular graphs of fixed degree

摘要： 我们研究具有大但固定度数$d$的随机$d$-正则图。在体谱$[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$中，我们证明了虽然其条目不集中，但格林函数可以准确地由仅依赖于图的局部结构的随机变量近似，直至最优谱尺度。这个稳定性结果意味着，其他结果之一是，对于谱密度的 Kesten--McKay 定律适用于最小可能的尺度，并且体特征向量完全扩展并满足量子唯一遍历性的强形式。我们的方法结合了随机正则图在小距离处的众所周知的树状（少环）结构和大距离处的随机矩阵行为。 显示英文摘要

摘要： We study random $d$-regular graphs with large but fixed degree $d$. In the bulk spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$, we prove that, while its entries do not concentrate, the Green's function is accurately approximated by random variables that only depend on the local structure of the graph, down to the optimal spectral scale. This stability result implies, among other consequences, that the Kesten--McKay law for the spectral density applies down to the smallest possible scale, and that the bulk eigenvectors are completely delocalized and satisfy a strong form of quantum unique ergodicity. Our method combines the well-known tree-like (few cycles) structure of random regular graphs at small distances with random matrix-like behavior at large distances.