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A Matrix Representation of the Bercovici-Pata Bijection
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Thierry Cabanal-Duvillard, Université Paris 5
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Abstract
Let μ be an infinitely divisible
law on the real line, Λ(μ) its freely infinitely
divisible image by the Bercovici-Pata bijection. The purpose of this article is to produce a new kind of
random matrices with distribution μ at dimension 1, and with its
empirical spectral law converging to Λ(μ) as the dimension tends to infinity. This constitutes a generalisation of Wigner's result
for the Gaussian Unitary Ensemble.
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Full text: PDF
Pages: 632-661
Published on: June 13, 2005
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Bibliography
TITLE HERE
| | Anshelevich, Michael. Partition-dependent stochastic measures and $q$-deformed cumulants.
Doc. Math. 6 (2001), 343--384 (electronic). MR1871667 (2004k:46107) |
| Arnold, Ludwig. On the asymptotic distribution of the eigenvalues of random matrices.
J. Math. Anal. Appl. 20 1967 262--268. MR0217833 (36 #922) |
| Barndorff-Nielsen, Ole E.; Thorbjørnsen, Steen. Self-decomposability and Lévy processes in free probability.
Bernoulli 8 (2002), no. 3, 323--366. MR1913111 (2003c:60031) |
| Barndorff-Nielsen, O. E.; Thorbjørnsen, S. A connection between free and classical infinite divisibility.
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004), no. 4, 573--590. MR2105912 |
| Benaych-Georges, Florent.
Classical and free infinitely divisible distributions and random
matrices. Preprint, (2004).
|
| Bercovici, Hari; Voiculescu, Dan. Free convolution of measures with unbounded support.
Indiana Univ. Math. J. 42 (1993), no. 3, 733--773. MR1254116 (95c:46109) |
| Bercovici, Hari; Pata, Vittorino. Stable laws and domains of attraction in free probability theory.
With an appendix by Philippe Biane.
Ann. of Math. (2) 149 (1999), no. 3, 1023--1060. MR1709310 (2000i:46061) |
| Bhatia, Rajendra. Matrix analysis.
Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. xii+347 pp. ISBN: 0-387-94846-5 MR1477662 (98i:15003) |
| Biane, Philippe. Processes with free increments.
Math. Z. 227 (1998), no. 1, 143--174. MR1605393 (99e:46085) |
| Collins, Benoît. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability.
Int. Math. Res. Not. 2003, no. 17, 953--982. MR1959915 (2003m:28015) |
| Good, I. J. A new formula for cumulants.
Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 2, 333--337. MR0386098 (52 #6957) |
| Groeneveld, J.; van Kampen, N. G. A simple proof of an identity of Good.
Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 2, 183--184. MR0438551 (55 #11462) |
| Guionnet, A.; Zeitouni, O. Concentration of the spectral measure for large matrices.
Electron. Comm. Probab. 5 (2000), 119--136 (electronic). MR1781846 (2001k:15035) |
| Hiai, Fumio; Petz, Dénes. The semicircle law, free random variables and entropy.
Mathematical Surveys and Monographs, 77. American Mathematical Society, Providence, RI, 2000. x+376 pp. ISBN: 0-8218-2081-8 MR1746976 (2001j:46099) |
| Kruglov, V.M. A note on infinitely divisible distributions.
Theory Probab. Appl., 15 (1970) 319--324.
MR0272018 (42 #6899) |
| Ledoux, Michel. The concentration of measure phenomenon.
Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9 MR1849347 (2003k:28019) |
| Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces.
Isoperimetry and processes.
Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9 MR1102015 (93c:60001) |
| Lehner, Franz. Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems.
Math. Z. 248 (2004), no. 1, 67--100. MR2092722 |
| Marçenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices.
(Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536. MR0208649 (34 #8458) |
| McCarthy, Charles A. $csb{p}$.
Israel J. Math. 5 1967 249--271. MR0225140 (37 #735) |
| Sato, Ken-iti. Lévy processes and infinitely divisible distributions.
Translated from the 1990 Japanese original.
Revised by the author.
Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4 MR1739520 (2003b:60064) |
| Shatashvili, Samson L. Correlation functions in the Itzykson-Zuber model.
Comm. Math. Phys. 154 (1993), no. 2, 421--432. MR1224086 (95e:81140) |
| Simon, Barry. Trace ideals and their applications.
London Mathematical Society Lecture Note Series, 35. Cambridge University Press, Cambridge-New York, 1979. viii+134 pp. ISBN: 0-521-22286-9 MR0541149 (80k:47048) |
| Voiculescu, Dan. Limit laws for random matrices and free products.
Invent. Math. 104 (1991), no. 1, 201--220. MR1094052 (92d:46163) |
| Wigner, Eugene P. On the distribution of the roots of certain symmetric matrices.
Ann. of Math. (2) 67 1958 325--327. MR0095527 (20 #2029) |
| Xu, Feng. A random matrix model from two-dimensional Yang-Mills theory.
Comm. Math. Phys. 190 (1997), no. 2, 287--307. MR1489573 (99f:81185) |
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