2004/06/30 by Florent Benaych-Georges
Computer Science · Mathematics · #Mathematical functions and polynomials #Matrix Theory and Algorithms #Random Matrices and Applications #math.OA #math.PR #msc:15A52 #msc:46L54 #msc:60E07 #msc:60F05.
paper · pdf · doi:10.1214/009117904000000982
published as Annals of Probability 2005, Vol. 33, No. 3, 1134-1170 · Published at http://dx.doi.org/10.1214/009117904000000982 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/05/01 · arxiv created 2005/08/30 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙdμ on the space of d×d Hermitian matrices such that ℙdμ*ℙdν=ℙdμ*ν. The spectral distribution of a random matrix with distribution ℙdμ converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko–Pastur distribution. In an analogous way, for every d≥1, we associate to each *-infinitely divisible distribution μ, a distribution \mathbbLdμ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |Md|, when Md is \mathbbLdμ-distributed, converges in probability to Ψ(μ).