Spectrum of random d-regular graphs up to the edge

IF 3.1 1区数学 Q1 MATHEMATICS Communications on Pure and Applied Mathematics Pub Date : 2023-09-28 DOI:10.1002/cpa.22176

Jiaoyang Huang, Horng-Tzer Yau

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Abstract

Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree $d ⩾ 3$ . We prove that, with probability $1 - N^{- 1 + ε}$ for any $ε > 0$ , the following two properties hold as $N \to \infty$ provided that $d ⩾ 3$ : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, $λ_{2}, | λ_{N} | ⩽ 2 + N^{- c}$ . (ii) All eigenvectors of random d-regular graphs are completely delocalized.

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随机 d 规则图谱，直至边缘

考虑 N 个顶点上具有固定度 d ⩾ 3 $d\geqslant 3$ 的随机 d-regular 图的归一化邻接矩阵。我们证明，对于任意 ε > 0 $\varepsilon >0$ ，只要 d ⩾ 3 $d\geqslant 3$，以下两个性质在 N → ∞ $N \rightarrow \infty$ 时成立：(i) 特征值接近凯斯顿-麦凯分布给出的经典特征值位置。特别是，极值特征值以 N 的多项式误差约束集中，即 λ 2 , | λ N | | ⩽ 2 + N - c $\lambda _2, |\lambda _N|leqslant 2+N^{-c}$ 。 (ii) 随机 d-regular 图形的所有特征向量都是完全非局部化的。

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来源期刊

Communications on Pure and Applied Mathematics 数学-数学

CiteScore

6.70

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Communications on Pure and Applied Mathematics (ISSN 0010-3640) is published monthly, one volume per year, by John Wiley & Sons, Inc. © 2019. The journal primarily publishes papers originating at or solicited by the Courant Institute of Mathematical Sciences. It features recent developments in applied mathematics, mathematical physics, and mathematical analysis. The topics include partial differential equations, computer science, and applied mathematics. CPAM is devoted to mathematical contributions to the sciences; both theoretical and applied papers, of original or expository type, are included.