Eigenvalues of random lifts and polynomials of random permutation matrices

Abstract

Let (σ 1 , . . . , σ d ) be 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 → ∞, these permutations viewed as operators on the n − 1 dimensional vector space {(x 1 , . . . , x n ) ∈ C n , x i = 0}, 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.

The spectral norm of random lifts of matrices

“…We note that the proof of Theorem 1.3 is not exploiting any potential structure of the "lifting matrices" Π ij . In fact, this may explain why Theorem 1.5 is worse than the result in [BC19] by an additive log(kn) factor in the large k limit for d-regular base graphs. One may be able to obtain a stronger result, for instance E A (k,π) − EA (k,π) ≤ 2 √ ∆ + o k (1), with a more careful analysis considering that Π ij are permutation matrices.…”

“…Acknowledgments. 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.…”

On a linearization trick

“…For any y ∈ S 2 we have 0 y y * 0 ∈ S 3 , and hence y = (1 0) 0 y y * 0 0 1 . This shows that a factorization of the form (5) with S 2 can be transformed into one with S 3 .…”

“…It was introduced in the Gaussian random matrix context by Haagerup and Thorbjørnsen [8], who mention in [8] that they were inspired by a similar trick from the author's [12]. The latter can be applied, among other settings, to unitary random matrices, in problems about "strong convergence" considered more recently by Collins and Male in [6], and Bordenave and Collins in [5]. Roughly, one wants to estimate the limit of the norm of a "polynomial" P (x , ...) in large unitary random N × N -matrices and their inverses when N → ∞ and to show that the limit is equal to the norm of the same polynomial P (x ∞ 1 , x ∞ 2 , ...; x ∞ 1 * , x ∞ 2 * , ...) but with the random matrices replaced by certain unitary matrices (x ∞ 1 , x ∞ 2 , ...) that play the role of a limiting object.…”

“…We assume (1) and again that K 0 ⊗ A is the algebra generated by (K 0 ⊗ 1 A ) ∪ S. Then, the following are equivalent: Remark 5. Assume (this is the main case of interest for us) that c = 1, and that S is stable by taking block diagonal sums of the form (2) with diagonal coefficients in S. Then the factorization in the preceding Corollary 4 can be stated just like this: Any x ∈ M n (A) with x < 1 can be written as a product (5) x = α 0 D 1 α 1 . .…”