Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 067, 24 pages      arXiv:1901.03117      https://doi.org/10.3842/SIGMA.2019.067

Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

Theodoros Assiotis
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

Received April 08, 2019, in final form September 04, 2019; Published online September 11, 2019

Abstract
The ergodic unitarily invariant measures on the space of infinite Hermitian matrices have been classified by Pickrell and Olshanski-Vershik. The much-studied complex inverse Wishart measures form a projective family, thus giving rise to a unitarily invariant measure on infinite positive-definite matrices. In this paper we completely solve the corresponding problem of ergodic decomposition for this measure.

Key words: infinite random matrices; ergodic measures; inverse Wishart measures; orthogonal polynomials.

pdf (496 kb)   tex (27 kb)  

References

  1. Baryshnikov Yu., GUEs and queues, Probab. Theory Related Fields 119 (2001), 256-274.
  2. Borodin A., Olshanski G., Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87-123, arXiv:math-ph/0010015.
  3. Bru M.-F., Wishart processes, J. Theoret. Probab. 4 (1991), 725-751.
  4. Bufetov A.I., Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures, Izv. Math. 79 (2015), 1111-1156, arXiv:1312.3161.
  5. Bufetov A.I., Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. II. Convergence of infinite determinantal measures, Izv. Math. 80 (2016), 299-315, arXiv:1610.04639.
  6. Bufetov A.I., Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures III. The infinite Bessel process as the limit of radial parts of finite-dimensional projections of infinite Pickrell measures, Izv. Math. 80 (2016), 1035-1056, arXiv:1610.04646.
  7. Bufetov A.I., Qiu Y., The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures, Ark. Mat. 54 (2016), 403-435, arXiv:1402.5230.
  8. Capitaine M., Casalis M., Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices, Indiana Univ. Math. J. 53 (2004), 397-431, arXiv:math.PR/0504414.
  9. Cunden F.D., Dahlqvist A., O'Connell N., Integer moments of complex Wishart matrices and Hurwitz numbers, arXiv:1809.10033.
  10. Cunden F.D., Mezzadri F., Simm N., Vivo P., Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, J. Phys. A: Math. Teor. 49 (2016), 18LT01, 20 pages, arXiv:1601.06690.
  11. Cunden F.D., Mezzadri F., Simm N., Vivo P., Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities, J. Math. Phys. 57 (2016), 111901, 16 pages, arXiv:1607.00250.
  12. Defosseux M., Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincar'e Probab. Stat. 46 (2010), 209-249, arXiv:0810.1011.
  13. Forrester P.J., The spectrum edge of random matrix ensembles, Nuclear Phys. B 402 (1993), 709-728.
  14. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  15. Gelfand I.M., Naimark M.A., Unitary representations of classical groups, Proceedings Steklov Mathematical Institute, Vol. 36, Izdat. Nauk SSSR, Moscow-Leningrad, 1950.
  16. Gorin V., Olshanski G., A quantization of the harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal. 270 (2016), 375-418, arXiv:1504.06832.
  17. Gupta A.K., Nagar D.K., Matrix variate distributions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2000.
  18. Hua L.K., Harmonic analysis of functions of several complex variables in the classical domains, Translations of Mathematical Monographs, Vol. 6, Amer. Math. Soc., Providence, R.I., 1979.
  19. Johnstone I.M., On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), 295-327.
  20. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  21. Krall H.L., Frink O., A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949), 100-115.
  22. Mingo J.A., Speicher R., Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces, J. Funct. Anal. 235 (2006), 226-270, arXiv:math.OA/0405191.
  23. Neretin Yu.A., Hua-type integrals over unitary groups and over projective limits of unitary groups, Duke Math. J. 114 (2002), 239-266, arXiv:math-ph/0010014.
  24. Olshanski G., Vershik A., Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, in Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 175, Amer. Math. Soc., Providence, RI, 1996, 137-175.
  25. Pickrell D., Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), 323-356.
  26. Pickrell D., Mackey analysis of infinite classical motion groups, Pacific J. Math. 150 (1991), 139-166.
  27. Qiu Y., Infinite random matrices & ergodic decomposition of finite and infinite Hua-Pickrell measures, Adv. Math. 308 (2017), 1209-1268, arXiv:1410.1167.
  28. Rider B., Valkó B., Matrix Dufresne identities, Int. Math. Res. Not. 2016 (2016), 174-218, arXiv:1409.1954.
  29. Soshnikov A., Determinantal random point fields, Russian Math. Surveys 55 (2000), 923-975, arXiv:math.PR/0002099.
  30. Soshnikov A., Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields, J. Statist. Phys. 100 (2000), 491-522, arXiv:math-ph/9907012.
  31. Szegö G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., New York, 1939.
  32. Tracy C.A., Widom H., Level spacing distributions and the Bessel kernel, Comm. Math. Phys. 161 (1994), 289-309, arXiv:hep-th/9304063.
  33. Vershik A.M., Description of invariant measures for the actions of some infinite-dimensional groups, Soviet Math. Dokl. 15 (1974), 1396-1400.
  34. Wishart J., The generalized product moment distribution in samples from a normal multivariate population, Biometrika 20A (1928), 32-52.

Previous article  Next article  Contents of Volume 15 (2019)