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Classical and free infinitely divisible distributions and random matrices. (English) Zbl 1076.15022

H. Bercovici and V. Pata [Ann. Math. (2) 149, No. 3, 1023–1060 (1999; Zbl 0945.46046)] showed the existence of a homeomorphism \(\Psi\) between the sets of \(*\)-infinitely divisible and \(\boxplus\)-infinitely divisible distributions. In the article under review, the author constructs a matricial model for the \(\boxplus\)-infinitely divisible distributions that provides a more constructive way to see the bijection \(\Psi\) .
A direct application of this model is to independently recover M. Anshelevich’s construction of a continuum between the classical convolution \(*\) and the free convolution \(\boxplus\) for infinitely divisible measures [Doc. Math., J. DMV 6, 343–384 (2001; Zbl 1010.46064)].
The author also shows that the matrix model constructed in the paper allows one to recover Wigner’s result. It also gives a new proof of the convergence of the spectral distribution of the Wishart matrix with parameter 1 to the Marchenko-Pastur distribution, and it exposes the Marchenko-Pastur distribution as the limit spectral distribution of independent rank-one projections.

MSC:

15B52 Random matrices (algebraic aspects)
46L54 Free probability and free operator algebras
60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
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References:

[1] Anshelevich, M. V. (2001). Partition-dependent stochastic measures and \(q\)-deformed cumulants. Doc. Math. 6 343–384. · Zbl 1010.46064
[2] Barndorff-Nielsen, O. E. and Thorbjørnsen, S. (2002). Selfdecomposability and Lévy processes in free probability. Bernoulli 3 323–366. · Zbl 1024.60022
[3] 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. · Zbl 1059.60017
[4] 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. · Zbl 0945.46046
[5] Bercovici, H. and Voiculescu, D. (1993). Free convolution of measures with unbounded supports. Indiana Univ. Math. J. 42 733–773. · Zbl 0806.46070
[6] 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.
[7] Gnedenko, V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables . Adisson–Wesley, Reading, MA. · Zbl 0056.36001
[8] 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. · Zbl 0984.46042
[9] Hiai, F. and Petz, D. (2000). The Semicircle Law , Free Random Variables , and Entropy . Amer. Math. Soc., Providence, RI. · Zbl 0955.46037
[10] Pastur, L. and Vasilchuk, V. (2000). On the law of addition of random matrices. Comm. Math. Phys. 214 249–286. · Zbl 1039.82020
[11] Petrov, V. V. (1995). Limit Theorems of Probability Theory . Clarendon Press, Oxford. · Zbl 0826.60001
[12] Speicher, R. (1994). Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298 611–628. · Zbl 0791.06010
[13] Speicher, R. (1999). Notes of my lectures on combinatorics of free probability. Available at www.mast.queensu.ca/\(\sim\)speicher.
[14] Voiculescu, D. V. (1991). Limit laws for random matrices and free products. Invent. Math. 104 201–220. · Zbl 0736.60007
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