The spectral norm of random lifts of matrices

Abstract

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 $A^{(k,\mathrm{\pi })}$ 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

$$\mathbb{E}\Vert A^{(k,\mathrm{\pi })}\Vert \lesssim \underset{i}{max}\sqrt{\sum _j{A}_{ij}^2}+\underset{ij}{max}|A_{ij}|\sqrt{log(kn)}.$$

This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative $\sqrt{logn}$ factor in the Non-Commutative Khintchine inequality can be removed.

As a direct application of our result, we prove an upper bound of $2(1+\mathit{ϵ})\sqrt{\mathrm{\Delta }}+O(\sqrt{log(kn)})$ 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 $O(\sqrt{\mathrm{\Delta }log(kn)})$ by Oliveira [Oli10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives $2\sqrt{\mathrm{\Delta }-1}+o(1)$ as $k\to \mathrm{\infty }$ for Δ-regular graph G.