Sanov-type large deviations in Schatten classes

Abstract

Denote by λ 1 (A), . . . , λ n (A) the eigenvalues of an (n × n)-matrix A. Let Z n be an (n × n)-matrix chosen uniformly at random from the matrix analogue to the classical ℓ n pball, defined as the set of all self-adjoint (n × n)-matrices satisfying n k=1 |λ k (A)| p ≤ 1. We prove a large deviations principle for the (random) spectral measure of the matrix n 1/p Z n . As a consequence, we obtain that the spectral measure of n 1/p Z n converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as n → ∞. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.

Yet another note on the arithmetic-geometric mean inequality

“…Then it can be easily checked that (10) and (11) hold. The uniqueness follows from the fact that (10) and (11) imply H s * + 1 p = pα − log(β) < 0,…”

“…While large deviations are extensively studied in probability theory (see, e.g., [4,5] and the references cited therein), they have not been considered -contrary to central limit theorems -in geometric functional analysis until the very recent paper by Gantert, Kim, and Ramanan [6]. Already shortly after, this work has been extended and complemented in [3,9,10,12,13]. In contrast to the universality in central limit theorems, the probabilities of (large) deviations on the scale of laws of large numbers, are non-universal, thus being sensitive to the distribution of the random variables considered.…”

Geometry of ℓ p n -Balls $$\ell _p^n\,\text{-Balls}$$ : Classical Results and Recent Developments

“…The rate function identified is essentially the so-called relative entropy perturbed by some p-th moment penalty (see [16,Equation (3.4)]). While this result is again in the commutative setting of the ℓ n p -balls, Kabluchko, Prochno, and Thäle [13] recently studied principles of large deviations in the non-commutative framework of self-adjoint and classical Schatten p-classes. The self-adjoint setting is the one of the classical matrix ensembles which has already been introduced in Subsection 4.2.2 (to avoid introducing further notation, for the case of Schatten trace classes we refer the reader to [13] directly).…”

“…While this result is again in the commutative setting of the ℓ n p -balls, Kabluchko, Prochno, and Thäle [13] recently studied principles of large deviations in the non-commutative framework of self-adjoint and classical Schatten p-classes. The self-adjoint setting is the one of the classical matrix ensembles which has already been introduced in Subsection 4.2.2 (to avoid introducing further notation, for the case of Schatten trace classes we refer the reader to [13] directly). In the spirit of [16], they proved a so-called Sanov-type large deviations principles for the spectral measure of n 1/p multiples of random matrices chosen uniformly (or with respect to the cone measure on the boundary) from the unit balls of self-adjoint and non self-adjoint Schatten pclasses where 0 < p ≤ +∞.…”

“…and this can be made explicit to get Equation (13). The remaining case of p = ∞ can be repeated using the aforementioned conventions on the quantities M ∞ and C ∞ .…”