The Annals of Probability

Classical and free infinitely divisible distributions and random matrices

Florent Benaych-Georges

Full-text: Open access

Abstract

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 $\mathbb{L}_{d}^{\mu}$ 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 $\mathbb{L}_{d}^{\mu}$-distributed, converges in probability to Ψ(μ).

Article information

Source
Ann. Probab., Volume 33, Number 3 (2005), 1134-1170.

Dates
First available in Project Euclid: 6 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1115386721

Digital Object Identifier
doi:10.1214/009117904000000982

Mathematical Reviews number (MathSciNet)
MR2135315

Zentralblatt MATH identifier
1076.15022

Subjects
Primary: 15A52 46L54: Free probability and free operator algebras
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems

Keywords
Random matrices free probability asymptotic freeness free convolution Marchenko–Pastur distribution infinitely divisible distributions

Citation

Benaych-Georges, Florent. Classical and free infinitely divisible distributions and random matrices. Ann. Probab. 33 (2005), no. 3, 1134--1170. doi:10.1214/009117904000000982. https://projecteuclid.org/euclid.aop/1115386721


Export citation

References

  • Anshelevich, M. V. (2001). Partition-dependent stochastic measures and $q$-deformed cumulants. Doc. Math. 6 343--384.
  • Barndorff-Nielsen, O. E. and Thorbjørnsen, S. (2002). Selfdecomposability and Lévy processes in free probability. Bernoulli 3 323--366.
  • Barndorff-Nielsen, O. E. and Thorbjørnsen, S. (2004). A connection between free and classical infinite divisibility. Inf. Dimens. Anal. Quantum Probab. Relat. Top. 7 573--590.
  • Bercovici, H., Pata, V., with an appendix by Biane, P. (1999). Stable laws and domains of attraction in free probability theory. Ann. of Math. 149 1023--1060.
  • Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded supports. Indiana Univ. Math. J. 42 733--773.
  • Cabanal-Duvillard, T. (2004). About a matricial representation of the Bercovici--Pata bijection: A Lévy processes approach. Preprint. Available at www.math-info.univ-paris5.fr/$\sim$cabanal/liste-publi.html.
  • Gnedenko, V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Adisson--Wesley, Reading, MA.
  • Haagerup, U. and Larsen, F. (2000). Brown's spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176 331--367.
  • Hiai, F. and Petz, D. (2000). The Semicircle Law, Free Random Variables, and Entropy. Amer. Math. Soc., Providence, RI.
  • Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249--286.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford.
  • Speicher, R. (1994). Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 611--628.
  • Speicher, R. (1999). Notes of my lectures on combinatorics of free probability. Available at www.mast.queensu.ca/$\sim$speicher.
  • Voiculescu, D. V. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201--220.