Concentration and convergence rates for spectral measures of random matrices

Abstract

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.

Eigenvalue Variance Bounds for Wigner and Covariance Random Matrices

“…This is an improvement of Meckes and Meckes' rate of convergence obtained in [16]. Note however that the distance studied in [16] is the expected 1-Wasserstein distance between L N and its mean instead of ρ sc .…”

“…Note however that the distance studied in [16] is the expected 1-Wasserstein distance between L N and its mean instead of ρ sc . The rate of convergence in 1-Wasserstein distance can be furthermore compared to the rate of convergence in Kolmogorov distance.…”

“…The rate of convergence in terms of W 1 , also called the Kantorovich-Rubinstein distance, was studied by Guionnet and Zeitouni in [13] and recently by Meckes and Meckes in [16]. In [16], the authors…”

Random matrices with prescribed eigenvalues and expectation values for random quantum states

“…The required mixed moments of entries of random orthogonal matrices can also be found in [5]. For the Poincaré inequality estimate, one must condition on the coset of SO (n) within O (n); for similar arguments, see, e.g., [30]. The required spectral gap estimate on SO (n) can be found in [40].…”

“…Suppose now that Λ n op = o(n). Then Λ n 2 4 ≤ Λ n HS √ Λ nop = nσ n Λ n op , and so by (29) Next, by (30), for each i, U → a ii is a 2 Λ op -Lipschitz function on U (n), and so (24) It is well known that E max 1≤i≤d g i ≥ c √ log d. If we assume now that Λ n op ≤ K √ n then choosing d = ⌊ √ n⌋ completes the proof.…”