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Unitarily invariant ergodic matrices and free probability

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Abstract

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.

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References

  1. G. Olshanski and A. Vershik, “Ergodic the unitarily invariant measures on the space of infinite Hermitian matrices,” in Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2 (Amer. Math. Soc., Providence, RI, 1996), Vol. 175, pp. 137–175.

    MathSciNet  Google Scholar 

  2. A.M. Vershik, “Description of invariant measures for the actions of some infinite-dimensional groups,” Dokl. Akad. Nauk SSSR 218 (4), 749–752 (1974) [SovietMath. Dokl. 15 (4), 1396–1400 (1974)].

    MathSciNet  Google Scholar 

  3. A. Borodin, A. Bufetov, and G. Olshanski, “Limit shapes for growing extreme characters of U(8),” Ann. Appl. Probab. 25 (4), 2339–2381 (2015).

    Article  MathSciNet  Google Scholar 

  4. A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, in London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2006), Vol. 335.

  5. D. Voiculescu, “Addition of certain non-commuting random variables,” J. Funct. Anal. 66 (3), 323–346 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Pastur and V. Vasilchuk, “On the law of addition of random matrices,” Comm. Math. Phys. 214 (2), 249–286 (2000).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    Al. I. Bufetov

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  1. Al. I. Bufetov
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Correspondence to Al. I. Bufetov.

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Original Russian Text © Al. I. Bufetov, 2015, published in Matematicheskie Zametki, 2015, Vol. 98, No. 6, pp. 824–831.

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Bufetov, A.I. Unitarily invariant ergodic matrices and free probability. Math Notes 98, 884–890 (2015). https://doi.org/10.1134/S0001434615110206

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  • DOI: https://doi.org/10.1134/S0001434615110206

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