Matrix Concentration Inequalities and Free Probability

Abstract

A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality of Lust-Piquard and Pisier, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices X = i g i A i where g i are independent standard Gaussian variables and A i are matrix coefficients. This bound exhibits a logarithmic dependence on dimension that is sharp when the matrices A i commute, but often proves to be suboptimal in the presence of noncommutativity.In this paper, we develop nonasymptotic bounds on the spectrum of arbitrary Gaussian random matrices that can capture noncommutativity. These bounds quantify the degree to which the deterministic matrices A i behave as though they are freely independent. This "intrinsic freeness" phenomenon provides a powerful tool for the study of various questions that are outside the reach of classical methods of random matrix theory. Our nonasymptotic bounds are easily applicable in concrete situations, and yield sharp results in examples where the noncommutative Khintchine inequality is suboptimal. When combined with a linearization argument, our bounds imply strong asymptotic freeness (in the sense of Haagerup-Thorbjørnsen) for a remarkably general class of Gaussian random matrix models, including matrices that may be very sparse and that lack any special symmetries.Beyond the Gaussian setting, we develop matrix concentration inequalities that capture noncommutativity for general sums of independent random matrices, which arise in many problems of pure and applied mathematics.

Strong convergence of tensor products of independent G.U.E. matrices

“…One defines (id m ⊗ id T ⊗ ν j ) applied to the resolvents in the left-hand sides as being the convergent infinite sums in the right-hand sides. 2 Let us establish next that formulae (35), (34) and (36) extend (when applied entrywise) to the above power series expansion of the resolvent of (25).…”

“…Recent works established results of this nature, dealing with matrices X M , where the dimension of the G.U.E. matrices Y i 's is M and M = O(N 1/4 ) in [14], M = O(N 1/3 ) in [5], and M = o(N/(log N) 3 ) in [2]. As mentioned for instance in [2], this does not suffice for the purpose of [8], which requires M = N (see [2,Proposition 9.3]).…”

“…matrices Y i 's is M and M = O(N 1/4 ) in [14], M = O(N 1/3 ) in [5], and M = o(N/(log N) 3 ) in [2]. As mentioned for instance in [2], this does not suffice for the purpose of [8], which requires M = N (see [2,Proposition 9.3]).…”

“…2ℑz|ℑζ|, so that in order to ensure that X − ζ < |z − ζ|, one needs only make sure that 2 . By fixing a ℑζ ≪ 0, one finds an open set of z's around our initial point on which X − ζ < |z − ζ| holds.…”

Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

“…However if we let M fluctuate with N , then the answer is much less straightforward. In the case where every X N i is a GUE random matrix, then the convergence of the norm was proved for M ≪ N 1/4 in [40], it was improved to M ≪ N 1/3 in [18], and to M ≪ N/ ln 3 (N ) in [4]. Those results were motivated by [29] a paper of Ben Hayes which proved that the strong convergence of the family (X N i ⊗ I N , I N ⊗ Y N j ) i,j when (Y N j ) j are also independent GUE random matrices of size N implies some rather important result on the structure of certain finite von Neumann algebras, the so-called Peterson-Thom conjecture.…”