Local Kesten-McKay Law for Random Regular Graphs
Local Kesten-McKay Law for Random Regular Graphs
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DOI：
10.1007/s00220-019-03345-3
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发表时间：
2019-07-01
影响因子：
2.4
通讯作者：
Yau, Horng-Tzer
中科院分区：
文献类型：
--
作者：
Bauerschmidt, Roland;Huang, Jiaoyang;Yau, Horng-Tzer
我们研究具有大但固定的度\(d\)的随机\(d\)-正则图的邻接矩阵。在从谱的主体部分\([-2\sqrt{d - 1}+\epsilon, 2\sqrt{d - 1}-\epsilon]\)一直到最优谱尺度，我们证明格林函数可以由某些仅依赖于原始图局部结构的无限树状（很少有圈）图的格林函数来逼近。这个结果意味着对于谱密度一直到最小尺度以及主体特征向量的完全离域化，凯斯滕 - 麦凯定律成立。我们的方法基于对邻接矩阵的格林函数的估计以及对图中大球的边界边的重采样。
We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum [-2d-1+epsilon,2d-1 epsilon] 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.