Open Access
2021 The spectral norm of random lifts of matrices
Afonso S. Bandeira, Yunzi Ding
Author Affiliations +
Electron. Commun. Probab. 26: 1-10 (2021). DOI: 10.1214/21-ECP415


We study the spectral norm of random lifts of matrices. Given an n×n symmetric matrix A, and a centered distribution π on k×k(k2) symmetric matrices with spectral norm at most 1, let the matrix random lift A(k,π) be the random symmetric kn×kn matrix (AijXij)1i<jn, where Xij are independent samples from π. We prove that

EA(k,π)max i j Aij2+maxij|Aij| 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 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+ϵ) Δ+O( 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( Δlog(kn)) by Oliveira [Oli10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives 2 Δ1+o(1) as k for Δ-regular graph G.

Funding Statement

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.


Download Citation

Afonso S. Bandeira. Yunzi Ding. "The spectral norm of random lifts of matrices." Electron. Commun. Probab. 26 1 - 10, 2021.


Received: 22 August 2020; Accepted: 28 June 2021; Published: 2021
First available in Project Euclid: 1 August 2021

arXiv: 2006.06505
Digital Object Identifier: 10.1214/21-ECP415

Primary: 60C99

Keywords: concentration inequality , matrix lifts , Random matrix theory

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