Regularization by Free Additive Convolution, Square and Rectangular Cases

Abstract

Abstract. The free convolution ⊞ is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution ⊞ λ allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups (µt) of probability measures such that µ0 = δ0 and such that for all t > 0 and all probability measure ν, µt⊞ν (or, in the rectangular context, µt⊞ λ ν) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, for ⊞, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for µ in a ⊞ λ -continuous semigroup, µ⊞ λ ν either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, and thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of µ⊞ λ ν except on a negligible set of points, as well as existence and continuity of a density everywhere.

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

“…Setting λ = 0, (2.2) reduces to 4) and one immediately checks that in this case µ f c is given, as expected, by the standard semicircular measure, µ sc , which is characterized by the density µ sc (E) = 1 2π…”

“…Similarly to (2.2), the free convolution measure ω 1 ⊞ ω 2 can be described in terms of a set of functional equations for the Stieltjes transforms; see [39,11,5]. For a discussion of regularity properties of ω 1 ⊞ ω 2 we refer to [4]. Free probability theory turned out to be a natural setting for studying global laws for such ensembles; see, e.g., [51,2].…”

“…However, if λ < λ + , then there are N -independent constants L + ≡ L + (µ, λ) and c ≡ c(µ, λ), such that 4) where Φ c denotes the cumulative distribution function of the centered Gaussian distribution with variance c; see Appendix C. We remark that neither (1.3) nor (1.4) depend on the symmetry type of the Wigner matrix W . The appearance of the Weibull distribution in the model (1.1) is indeed expected when λ grows sufficiently fast with N , since in this case the diagonal matrix dominates the spectral properties of H. However, it is quite surprising that the Weibull distribution already appears for λ order one, since the local behavior of the eigenvalues in the bulk of the deformed model mainly stems from the Wigner part, and the contribution from the random diagonal part is limited to mesoscopic fluctuations of the eigenvalues; see [32].…”

Local single ring theorem

“…The problem here is that the upper-bound on E 1 N Tr G H (z) − m µ s ν s (z) deduced from Equations (3)-(6) involves the inverse of a certain 2 × 2 determinant (see (31)), which can vanish for z close to the real line (to control this determinant for z close to the real line, one would need precise informations about the density of ν ∞,z , which have, except for the case of i.i.d. matrices, remained out of reach so far, despite several studies of these questions as in [5,7,9]) 1 . However, using only some bounds on the operator norms of A and B, we can deduce from (3)-(6) that for |z| large enough,…”

Rectangular random matrices, related convolution

“…Firstly, in [4], we study the related infinite divisibility: it is proved that the set of λ -infinitely divisible distributions is in a deep correspondence with the set of symmetric classical infinitely divisible distributions. Secondly, in [3], we study some questions related to the support and the regularity of µ 1 λ µ 2 .…”