The spectral norm of random lifts of matrices

We study the spectral norm of matrix random lifts $A^{(k,\pi)}$ for a given $n\times n$ matrix $A$ and $k\ge 2$, which is a random symmetric $kn\times kn$ matrix whose $k\times k$ blocks are obtained by multiplying $A_{ij}$ by a $k\times k$ matrix drawn independently from a distribution $\pi$ supported on $k\times k$ matrices with spectral norm at most $1$. Assuming that $\mathbb{E}_\pi X = 0$, we prove that \[\mathbb{E} \|A^{(k,\pi)}\|\lesssim \max_{i}\sqrt{\sum_j A_{ij}^2}+\max_{ij}|A_{ij}|\sqrt{\log (kn)}.\] 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. We also show an application on random $k$-lifts of graphs (each vertex of the graph is replaced with $k$ vertices, and each edge is replaced with a random bipartite matching between the two sets of $k$ vertices each). We prove an upper bound of $2(1+\epsilon)\sqrt{\Delta}+O(\sqrt{\log(kn)})$ on the new eigenvalues for random $k$-lifts of a fixed $G = (V,E)$ with $|V| = n$ and maximum degree $\Delta$, compared to the previous result of $O(\sqrt{\Delta\log(kn)})$ by Oliveira [Oli09] and the recent breakthrough by Bordenave and Collins [BC19] which gives $2\sqrt{\Delta-1} + o(1)$ as $k\rightarrow\infty$ for $\Delta$-regular graph $G$.