On the free convolution with a semi-circular distribution. (English) Zbl 0904.46045

Biane, Philippe

Indiana Univ. Math. J. 46, No. 3, 705-718 (1997).

A formula for the density of the free convolution of an arbitrary probability measure with a semi-circular distribution is given. The semi-circular distribution is the analogue in free probability theory of the Gaussian distribution, so that what is obtained is an explicit formula for the solution to the free analogue of the heat equation, which can be interpreted as the limit, for large \(N\), of the heat equation on the space of \(N\times N\) Hermitian matrices. This formula is used to establish a certain number of regularity properties of the measures obtained in this way. In particular, it is proved that the measures obtained by free convolution with a semi-circular distribution of variance \(t\) have always a continuous density whose cube is Lipschitz, with Lipschitz constant less than \(3(4\pi^3t^2)^{-1}\). Also, an optimal bound on the density of such measures, when the starting measure has a bounded density, is given.

Reviewer: Ph.Biane (Paris)

MSC:

46L51	Noncommutative measure and integration
46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras
30C20	Conformal mappings of special domains

Keywords:

conformal mapping; free convolution; semi-circular distribution; free probability theory; heat equation