Intersection of unit balls in classical matrix ensembles

Sanov-type large deviations in Schatten classes

“…As we shall see, the good rate function governing the LDPs is essentially given by the logarithmic energy (which is remarkably the same as the negative of Voiculescu's free entropy introduced in [34]) and, which is quite interesting, a perturbation by a constant, which is strongly connected to the famous Ullman distribution. In fact, this constant already appeared in our recent works [21,22], where the precise asymptotic volume of unit balls in classical matrix ensembles and Schatten trace classes were computed using ideas from the theory of logarithmic potentials with external fields. As a consequence of our LDPs, we obtain a law of large numbers and show that the spectral measure converges weakly almost surely to the Ullman distribution, as the dimension tends to infinity (see Corollary 1.4).…”

“…For example, it were Gordon and Lewis [12] who obtained that the space S 1 does not have local unconditional structure, Tomzcak-Jaegermann [33] demonstrated that this space (naturally identified with the projective tensor product ℓ 2 ⊗ π ℓ 2 ) has Rademacher cotype 2, Szarek and Tomczak-Jaegermann [32] provided bounds for the volume ratio of S n 1 , and König, Meyer and Pajor [26] proved the boundedness of the isotropic constants of S n p (1 ≤ p ≤ +∞). More recently, Guédon and Paouris [15] have established concentration of mass properties for the unit balls of Schatten p-classes and classical matrix ensembles, Barthe and Cordero-Erausquin [6] studied variance estimates, Chávez-Domínguez and Kutzarova determined the Gelfand widths of certain identity mappings between finite-dimensional trace classes S p , Radke and Vritsiou [28] and Vritsiou [35] proved the thin-shell conjecture and the variance conjecture for the operator norm, respectively, Hinrichs, Prochno and Vybíral [19] computed the entropy numbers for natural embeddings of S n p in all possible regimes, and Kabluchko, Prochno and Thäle [21,22] obtained the precise asymptotic volumes of the unit balls in classical matrix ensembles and Schatten classes, studied volumes of intersections (in the spirit of Schechtman and Schmuckenschläger [30]) and determined the exact asymptotic volume ratios for S n p (0 < p ≤ +∞). It can be seen from all this work referenced above that while those matrix spaces often show a certain similarity to the commutative setting of classical ℓ n p -spaces, there is a considerable difference in the behavior of certain quantities related to the geometry of Banach spaces.…”

“…However, in our case, we need to control the deviations of the empirical measures and, in addition, their p-th moments towards arbitrary small balls in the product topology of the weak topology on the space of probability measures and the standard topology on R in the self-adjoint or R + in the non self-adjoint set-up, respectively, and then prove exponential tightness. We shall also use a probabilistic representation for random points in the unit balls of classical matrix ensembles obtained recently in [22] (see ( 4) and ( 6)) and a non self-adjoint counterpart (see Proposition 8.2). As we shall see, the good rate function governing the LDPs is essentially given by the logarithmic energy (which is remarkably the same as the negative of Voiculescu's free entropy introduced in [34]) and, which is quite interesting, a perturbation by a constant, which is strongly connected to the famous Ullman distribution.…”

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

“…For instance, Carl and Defant [7] proved a Garnaev-Gluskin result for Gelfand numbers of Schatten class embeddings S N p → S N 2 for 1 ≤ p ≤ 2, König, Meyer, and Pajor [30] obtained that the isotropic constants of S N p unit balls are bounded above by absolute constants for all 1 ≤ p ≤ ∞, and Guédon and Paouris studied their concentration of mass properties in [19]. Even more recently, Radke and Vritsiou succeeded in confirming the thin-shell conjecture for S N ∞ [35], Hinrichs, Prochno, and Vy-bíral computed the entropy numbers for identity mappings S N p → S N q for all 0 < p, q ≤ ∞ [23] and recently obtained asymptotically sharp estimates for Gelfand numbers in almost all regimes [24], Vritsiou confirmed the variance conjecture for S N ∞ [44], and, in a series of papers, Kabluchko, Prochno, and Thäle computed the exact asymptotic volume and volume ratio of S N p unit balls for 0 < p ≤ ∞ [26], studied the threshold behavior of the volume of intersections of unit balls [27], and obtained large deviation principles for the empirical spectral measures of random matrices in Schatten unit balls [28].…”

The maximum entropy principle and volumetric properties of Orlicz balls

“…Applications of those results include an asymptotic version of a result of Schechtman and Zinn [15, Subsection 2.5], a demonstration that in the critical case arbitrary limits in (0, 1) can occur [15, Corollary 2.2], a result on the volume of intersections of neighboring and multiple balls [15,Corollary 2.3], where the answer in the critical case is not 2 −d as may be expected, a comparison of random and non-random projections of ℓ d p -balls to lowerdimensional subspaces [16,Section 2], and several other applications. A non-commutative version of the Schechtman-Schmuckenschläger result for unit balls in classical random matrix ensembles was recently proved by Kabluchko, Prochno, and Thäle in [14]. We also refer the reader to the recent survey [26].…”