We study the spectral norm of random lifts of matrices. Given an symmetric matrix A, and a centered distribution π on symmetric matrices with spectral norm at most 1, let the matrix random lift be the random symmetric matrix , where are independent samples from π. We prove that
This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative factor in the Non-Commutative Khintchine inequality can be removed.
As a direct application of our result, we prove an upper bound of on the new eigenvalues for random k-lifts of a fixed with and maximum degree Δ, compared to the previous result of by Oliveira [Oli10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives as for Δ-regular graph G.
Afonso S. Bandeira partially supported by NSF grants DMS-1712730 and DMS-1719545, and by a grant from the Sloan Foundation. Yunzi Ding partially supported by NSF grant DMS-1712730.
We are grateful to Ramon van Handel for comments on an early version of the paper, in particular for pointing us to the latest results on graph k-lifts in [BC19], for directing us to the proof of Theorem 4.8 in [LvHY18] which allowed us to improve the constant factor before σ in Theorem 1.3 to 2, and for making us aware of recent efforts to improve the NCK inequality under more general settings. We would also like to thank Jiedong Jiang, Eyal Lubetzky, Ruedi Suter and Joel Tropp for helpful discussions.
"The spectral norm of random lifts of matrices." Electron. Commun. Probab. 26 1 - 10, 2021. https://doi.org/10.1214/21-ECP415