Eigenvalues of random lifts and polynomials of random permutation matrices

TL;DR

This paper investigates the spectral properties of random permutation matrices and their lifts, demonstrating asymptotic freeness, eigenvalue bounds, and applications to quantum expanders through advanced matrix and trace techniques.

Contribution

It develops a matrix-based non-backtracking operator theory and refines trace methods to analyze eigenvalues of random permutation matrices and their tensor products, extending results to quantum expanders.

Findings

Random permutations are asymptotically strongly free as n→∞.

Eigenvalues of random n-lifts approach the Alon-Boppana bound.

Schreier graphs generated by random permutations are close to Ramanujan.

Abstract

Consider a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n goes to infinity, these permutations viewed as operators on the (n-1) dimensional vector space orthogonal to the vector with all coordinates equal to 1, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method. As a byproduct, we show that the non-trivial eigenvalues of random n-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large n limit. This result generalizes Friedman's Theorem stating that with high probability, the Schreier graph generated by a finite number of independent random permutations is close to Ramanujan. Finally, we extend our results to tensor products of random permutation matrices. This extension is especially relevant in the context of quantum expanders.