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

SIGMA 19 (2023), 100, 18 pages      arXiv:2306.16323      https://doi.org/10.3842/SIGMA.2023.100
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Jacobi Beta Ensemble and $b$-Hurwitz Numbers

Giulio Ruzza ab
a) Grupo de Física Matemática, Campo Grande Edifício C6, 1749-016, Lisboa, Portugal
b) Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Campo Grande Edifício C6, 1749-016, Lisboa, Portugal

Received July 03, 2023, in final form November 29, 2023; Published online December 19, 2023

Abstract
We express correlators of the Jacobi $\beta$ ensemble in terms of (a special case of) $b$-Hurwitz numbers, a deformation of Hurwitz numbers recently introduced by Chapuy and Dołęga. The proof relies on Kadell's generalization of the Selberg integral. The Laguerre limit is also considered. All the relevant $b$-Hurwitz numbers are interpreted (following Bonzom, Chapuy, and Dołęga) in terms of colored monotone Hurwitz maps.

Key words: beta ensembles; Jack polynomials; Hurwitz numbers; combinatorial maps.

pdf (503 kb)   tex (27 kb)  

References

  1. Ambjørn J., Makeenko Yu.M., Properties of loop equations for the Hermitian matrix model and for two-dimensional quantum gravity, Modern Phys. Lett. A 5 (1990), 1753-1763.
  2. Bertola M., Harnad J., Rationally weighted Hurwitz numbers, Meijer $G$-functions and matrix integrals, J. Math. Phys. 60 (2019), 103504, 15 pages, arXiv:1904.03770.
  3. Bonzom V., Chapuy G., Dołęga M., $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral, Int. Math. Res. Not. 2023 (2023), 12172-12230, arXiv:2109.01499.
  4. Borot G., Eynard B., Mulase M., Safnuk B., A matrix model for simple Hurwitz numbers, and topological recursion, J. Geom. Phys. 61 (2011), 522-540, arXiv:0906.1206.
  5. Borot G., Guionnet A., Asymptotic expansion of $\beta$ matrix models in the one-cut regime, Comm. Math. Phys. 317 (2013), 447-483, arXiv:1107.1167.
  6. Chapuy G., Dołęga M., Non-orientable branched coverings, $b$-Hurwitz numbers, and positivity for multiparametric Jack expansions, Adv. Math. 409 (2022), 108645, 72 pages, arXiv:2004.07824.
  7. Chekhov L., Eynard B., Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys. 2006 (2006), no. 12, 026, 29 pages, arXiv:math-ph/0604014.
  8. Cunden F.D., Dahlqvist A., O'Connell N., Integer moments of complex Wishart matrices and Hurwitz numbers, Ann. Inst. Henri Poincaré D 8 (2021), 243-268, arXiv:1809.10033.
  9. Deift P.A., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, 1999.
  10. Dijkgraaf R., Mirror symmetry and elliptic curves, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser, Boston, MA, 1995, 149-163.
  11. Dumitriu I., Edelman A., Matrix models for beta ensembles, J. Math. Phys. 43 (2002), 5830-5847, arXiv:math-ph/0206043.
  12. Edelman A., Sutton B.D., The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems, Found. Comput. Math. 8 (2008), 259-285.
  13. Ekedahl T., Lando S., Shapiro M., Vainshtein A., Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001), 297-327, arXiv:math.AG/0004096.
  14. Etingof P., Golberg O., Hensel S., Liu T., Schwendner A., Vaintrob D., Yudovina E., Introduction to representation theory (with historical interludes by Slava Gerovitch), Stud. Math. Libr., Vol. 59, American Mathematical Society, Providence, RI, 2011.
  15. Forrester P.J., Log-gases and random matrices, London Math. Soc. Monogr. Ser., Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  16. Forrester P.J., Rahman A.A., Witte N.S., Large $N$ expansions for the Laguerre and Jacobi $\beta$-ensembles from the loop equations, J. Math. Phys. 58 (2017), 113303, 25 pages, arXiv:1707.04842.
  17. Gisonni M., Grava T., Ruzza G., Laguerre ensemble: correlators, Hurwitz numbers and Hodge integrals, Ann. Henri Poincaré 21 (2020), 3285-3339, arXiv:1912.00525.
  18. Gisonni M., Grava T., Ruzza G., Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Lett. Math. Phys. 111 (2021), 67, 38 pages, arXiv:2011.04082.
  19. Goulden I.P., Guay-Paquet M., Novak J., Monotone Hurwitz numbers and the HCIZ integral, Ann. Math. Blaise Pascal 21 (2014), 71-89, arXiv:1107.1015.
  20. Goulden I.P., Jackson D.M., Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996), 873-892.
  21. Graczyk P., Letac G., Massam H., The complex Wishart distribution and the symmetric group, Ann. Statist. 31 (2003), 287-309.
  22. Graczyk P., Letac G., Massam H., The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution, J. Theoret. Probab. 18 (2005), 1-42.
  23. Guay-Paquet M., Harnad J., 2D Toda $\tau$-functions as combinatorial generating functions, Lett. Math. Phys. 105 (2015), 827-852, arXiv:1405.6303.
  24. Hanlon P.J., Stanley R.P., Stembridge J.R., Some combinatorial aspects of the spectra of normally distributed random matrices, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, American Mathematical Society, Providence, RI, 1992, 151-174.
  25. Harnad J., Orlov A.Yu., Hypergeometric $\tau$-functions, Hurwitz numbers and enumeration of paths, Comm. Math. Phys. 338 (2015), 267-284, arXiv:1407.7800.
  26. Hurwitz A., Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1-60.
  27. Itzykson C., Zuber J.-B., Matrix integration and combinatorics of modular groups, Comm. Math. Phys. 134 (1990), 197-207.
  28. Kadell K.W.J., The Selberg-Jack symmetric functions, Adv. Math. 130 (1997), 33-102.
  29. Killip R., Nenciu I., Matrix models for circular ensembles, Int. Math. Res. Not. 2004 (2004), 2665-2701, arXiv:math.SP/0410034.
  30. Macdonald I.G., Commuting differential operators and zonal spherical functions, in Algebraic Groups Utrecht 1986, Lecture Notes in Math., Vol. 1271, Springer, Berlin, 1987, 189-200.
  31. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Math. Monogr., Oxford University Press, New York, 1995.
  32. Novak J., On the complex asymptotics of the HCIZ and BGW integrals, arXiv:2006.04304.
  33. Okounkov A., Toda equations for Hurwitz numbers, Math. Res. Lett. 7 (2000), 447-453, arXiv:math.AG/0004128.
  34. Okounkov A., Pandharipande R., Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. 163 (2006), 517-560, arXiv:math.AG/0204305.
  35. Selberg A., Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71-78.
  36. Stanley R.P., Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115.

Previous article  Next article  Contents of Volume 19 (2023)