The Circular Law for random regular digraphs
The Circular Law for random regular digraphs
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随机正则有向图的循环定律
DOI：
10.1214/18-aihp943
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Nicholas A. Cook
中科院分区：
文献类型：
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作者：
Nicholas A. Cook
设$\log^Cn\le d\le n/2$，其中C>0$是一个充分大的常数，$A_n$表示n$个顶点上的均匀随机d$-正则有向图的邻接矩阵。我们证明，作为$n$趋于无穷大，经验谱分布的$A_n$，适当地重新标度，是由圆形法律。关键的一步是获得$A_n$的可加扰动的最小奇异值的定量的尾下界。
Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral distribution of $A_n$, suitably rescaled, is governed by the Circular Law. A key step is to obtain quantitative lower tail bounds for the smallest singular value of additive perturbations of $A_n$.