论文标题
矩阵随机升降的光谱规范
The Spectral Norm of Random Lifts of Matrices
论文作者
论文摘要
我们研究矩阵随机升力的光谱规范。给定$ n \ times n $对称矩阵$ a $，以及$ k \ times k \（k \ ge 2）$ k \ ge 2）$ symmetric矩阵，具有频谱标准$ 1 $，让矩阵随机升降$ a^{（k，π）$是随机的symmetric symmetric $ kn $ matrix， $（a_ {ij} x_ {ij}）_ {1 \ le i <j \ le n} $，其中$ x_ {ij {ij} $是$π$的独立样本。我们证明了这一点 $ \ MATHBB { 可以将此结果视为具有独立条目的随机矩阵上现有光谱界限的扩展，从而提供了进一步的实例，在非交通khintchine不平等中，乘法$ \ sqrt {\ log n} $因子可以删除。 作为我们结果的直接应用，我们证明了$ 2（1+ε）\sqrtδ+o（\ sqrt {\ sqrt {\ log（kn）}）$在新的eigenvalues上，用于随机$ k $ - 固定$ g =（v，e）$的随机$ k $ -lifts $ | v | v | v | v | v | = n $和最高度$δ$，与Oliveira的先前结果相比，Bordenave和Collins的最新突破是$ O（\ sqrt {Δ\ log（kn）}）$，它的突破给出了$ 2 \ sqrt {δ-1} + o（1）$（1）$（1）$ as $ k \ rightarrow for $ k \ rightarrow \ for $ $ $ $ $ $ - $ $ $ - $ - $ nreg- $ nreg- $ negrim-rengult
We study the spectral norm of random lifts of matrices. Given an $n\times n$ symmetric matrix $A$, and a centered distribution $π$ on $k\times k\ (k\ge 2)$ symmetric matrices with spectral norm at most $1$, let the matrix random lift $A^{(k,π)}$ be the random symmetric $kn\times kn$ matrix $(A_{ij}X_{ij})_{1\le i < j \le n}$, where $X_{ij}$ are independent samples from $π$. We prove that $$\mathbb{E} \|A^{(k,π)}\|\lesssim \max_{i}\sqrt{\sum_j A_{ij}^2}+\max_{ij}|A_{ij}|\sqrt{\log (kn)}.$$ This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative $\sqrt{\log n}$ factor in the Non-Commutative Khintchine inequality can be removed. As a direct application of our result, we prove an upper bound of $2(1+ε)\sqrtΔ+O(\sqrt{\log(kn)})$ on the new eigenvalues for random $k$-lifts of a fixed $G = (V,E)$ with $|V| = n$ and maximum degree $Δ$, compared to the previous result of $O(\sqrt{Δ\log(kn)})$ by Oliveira and the recent breakthrough by Bordenave and Collins which gives $2\sqrt{Δ-1} + o(1)$ as $k\rightarrow\infty$ for $Δ$-regular graph $G$.
