Regularization by free additive convolution, square and rectangular cases
Résumé
The free convolution (resp. its rectangular analogue) is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral (resp. singular) distributions, the spectral (resp. singular) distribution of a sum of independent unitarily invariant square (resp. rectangular) random matrices. In this paper, we consider these free convolutions, and study the possibility to find probability measures close to the Dirac mass at zero with regularization properties on the whole real line. More specifically, we try to find continuous semigroups $(\mu_t)$ of probability measures such that $\mu_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $\nu$, the free convolution of $\mu_t$ with $\nu$ (or, in the rectangular context, the rectangular free convolution of $\mu_t$ with $\nu$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, 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 $\mu$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $\mu$ with $\nu$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $\mu$ with $\nu$ except on a negligible set of points, as well as existence and continuity of a density everywhere.
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