随机 d 规则图谱，直至边缘

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC ACS Applied Electronic Materials Pub Date : 2023-09-28 DOI:10.1002/cpa.22176

Jiaoyang Huang, Horng-Tzer Yau

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引用次数: 0

摘要

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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Spectrum of random d-regular graphs up to the edge

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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来源期刊

ACS Applied Electronic Materials Multiple-

CiteScore

7.20

自引率

4.30%

发文量

567