Geodesic
Unitarily Invariant Ergodic Matrices and Free Probability
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 824-831.

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Probability measures on the space of Hermitian matrices which are ergodic for the conjugation action of an infinite-dimensional unitary group are considered. It is established that the eigenvalues of random matrices distributed with respect to these measures satisfy the law of large numbers. The relationship between such models of random matrices and objects in free probability, freely infinitely divisible measures, is also established.
Keywords: unitarily invariant ergodic matrix, infinitely divisible measure, free probability, Hermitian matrix, empiric distribution of eigenvalues, free convolution, free cumulant. 
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Al. I. Bufetov. Unitarily Invariant Ergodic Matrices and Free Probability. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 824-831. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a2/

[1] G. Olshanski, A. Vershik, “Ergodic unitarily invariant measures on the space of infinite Hermitian matrices”, Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996, 137–175 | MR | Zbl

[2] A. M. Vershik, “Opisanie invariantnykh mer dlya deistviya nekotorykh beskonechnomernykh grupp”, Dokl. AN SSSR, 218:4 (1974), 749–752 | MR | Zbl

[3] A. Borodin, A. Bufetov, G. Olshanski, “Limit shapes for growing extreme characters of $U(\infty)$”, Ann. Appl. Probab., 25:4 (2015), 2339–2381 | DOI | MR | Zbl

[4] A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability, London Math. Soc. Lecture Note Ser., 335, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl

[5] D. Voiculescu, “Addition of certain non-commuting random variables”, J. Funct. Anal., 66:3 (1986), 323–346 | DOI | MR | Zbl

[6] L. Pastur, V. Vasilchuk, “On the law of addition of random matrices”, Comm. Math. Phys., 214:2 (2000), 249–286 | DOI | MR | Zbl

[7] I. J. Schoenberg, “On Pólya frequency functions. I. The totally positive functions and their Laplace transforms”, J. Anal. Math., 1 (1951), 331–374 | DOI | MR | Zbl