标题： 局部Kesten--McKay定律对于随机正则图 显示英文标题

标题： Local Kesten--McKay law for random regular graphs

摘要： 我们研究了具有大但固定度数$d$的随机$d$-正则图的邻接矩阵。在谱的主体部分$[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$到最优谱尺度，我们证明了格林函数可以被某些仅依赖于原图局部结构的类似树状（少环）图的格林函数近似。这个结果意味着凯斯滕-麦凯定律适用于谱密度到最小尺度，并且体区特征向量完全扩展。我们的方法基于估计邻接矩阵的格林函数以及对图中大球边界边的重采样。 显示英文摘要

摘要： We study the adjacency matrices of random $d$-regular graphs with large but fixed degree $d$. In the bulk of the spectrum $[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]$ down to the optimal spectral scale, we prove that the Green's functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original graphs. This result implies that the Kesten--McKay law holds for the spectral density down to the smallest scale and the complete delocalization of bulk eigenvectors. Our method is based on estimating the Green's function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs.