## 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

### Citations:

Zbl 0945.46046; Zbl 1010.46064
Full Text:

### References:

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