Abstract
We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1.
Highlights
1.1 Beta and beta prime distributionsWigner’s semicircle law w and the Marchenko-Pastur law m, defined by √ 4 − x2 1 4−x w(dx) =1[−2,2](x) dx, m(dx) = 2π x 1[0,4](x) dx, are the most important distributions in free probability because they are respectively the limit distributions of the free central limit theorem and free Poisson’s law of small numbers
In the context of random matrices, w and m are the large N limit of the eigenvalue distributions of XN and XN2 respectively, where XN is an N × N normalized Wigner matrix. Those measures belong to the class of freely infinitely divisible distributions, the main subject of this paper. This class appears as the spectral distributions of large random matrices [BG05, C05]
We will clarify what property of the Cauchy transform in addition to a first-order differential equation guarantees free infinite divisibility
Summary
Wigner’s semicircle law w and the Marchenko-Pastur law (or free Poisson law) m, defined by. In the context of random matrices, w and m are the large N limit of the eigenvalue distributions of XN and XN2 respectively, where XN is an N × N normalized Wigner matrix Those measures belong to the class of freely infinitely divisible (or FID for short) distributions, the main subject of this paper. The other motivation of the present paper is to understand the result of [BBLS11] better, i.e. to understand the relationship between a first-order differential equation of the Cauchy transform and free infinite divisibility. The Cauchy transforms of distributions βp,q, βp,q, up, γp, γp−1, tq are all Gauss hypergeometric functions or limits of such and satisfy first-order differential equations. We will clarify what property of the Cauchy transform in addition to a first-order differential equation guarantees free infinite divisibility
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