矩阵随机提升的谱范数
The Spectral Norm of Random Lifts of Matrices
摘要：
研究了矩阵随机提升的谱范数。给定一个$n\times n$对称矩阵$A$，以及一个在$k\times k\ (k\ge 2)$对称矩阵上的中心分布$π$，其谱范数至多为$1$，令矩阵随机提升$A^{(k,π)}$为随机对称$kn\times kn$矩阵$(A_{ij}X_{ij})_{1\le i < j \le n}$，其中$X_{ij}$是来自$π$的独立样本。我们证明了$$\mathbb{E} \|A^{(k,π)}\|\lesssim \max_{i}\sqrt{\sum_j A_{ij}^2}+\max_{ij}|A_{ij}|\sqrt{\log (kn)}.$$这个结果可以被看作是现有的谱界的独立项的随机矩阵的扩展，提供了进一步的情况下，乘法$\sqrt{\log n}$因子的非交换Khintchine不等式可以删除。作为我们的结果的一个直接应用，我们证明了一个新的特征值的随机$k$ -升降机的固定$G = (V,E)$与$|V| = n$和最大程度$Δ$的$2(1+ε)\sqrtΔ+O(\sqrt{\log(kn)})$的上界，相比以前的结果$O(\sqrt{Δ\log(kn)})$ Oliveira和最近的突破，由Bordenave和柯林斯这给$2\sqrt{Δ-1} + o(1)$作为$k\rightarrow\infty$的$Δ$ -正则图$G$。
英文摘要：
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 $G$.
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