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

SIGMA 20 (2024), 054, 38 pages      arXiv:2312.12229      https://doi.org/10.3842/SIGMA.2024.054

Fay Identities of Pfaffian Type for Hyperelliptic Curves

Gaëtan Borot a and Thomas Buc-d'Alché b
a) Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
b) UMPA UMR 5669, ENS de Lyon, CNRS, 46, allée d'Italie 69007, Lyon, France

Received January 30, 2024, in final form June 06, 2024; Published online June 23, 2024

Abstract
We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation values of ratios of characteristic polynomials in ensembles of orthogonal or quaternionic self-dual random matrices. We show that they amount to identities for the theta function with the period matrix of a hyperelliptic curve, and in this form we reprove them by direct geometric methods.

Key words: random matrix theory; theta function; Fay's identity; hyperelliptic curves.

pdf (739 kb)   tex (66 kb)  

References

  1. Albeverio S., Pastur L., Shcherbina M., On the $1/n$ expansion for some unitary invariant ensembles of random matrices, Comm. Math. Phys. 224 (2001), 271-305.
  2. Anderson G.W., Guionnet A., Zeitouni O., An introduction to random matrices, Cambridge Stud. Adv. Math., Vol. 118, Cambridge University Press, Cambridge, 2009.
  3. Bertola M., Riemann surfaces and theta functions, Course at Concordia University, 2006, https://mypage.concordia.ca/mathstat/bertola/ThetaCourse/ThetaCourse.pdf.
  4. Bertola M., Free energy of the two-matrix model/dToda tau-function, Nuclear Phys. B 669 (2003), 435-461, arXiv:hep-th/0306184.
  5. Bertola M., Boutroux curves with external field: equilibrium measures without a variational problem, Anal. Math. Phys. 1 (2011), 167-211, arXiv:0705.3062.
  6. Bonnet G., David F., Eynard B., Breakdown of universality in multi-cut matrix models, J. Phys. A 33 (2000), 6739-6768, arXiv:cond-mat/0003324.
  7. Borodin A., Strahov E., Averages of characteristic polynomials in random matrix theory, Comm. Pure Appl. Math. 59 (2006), 161-253, arXiv:math-ph/0407065.
  8. Borot G., Eynard B., Geometry of spectral curves and all order dispersive integrable system, SIGMA 8 (2012), 100, 53 pages, arXiv:1110.4936.
  9. Borot G., Eynard B., Orantin N., Abstract loop equations, topological recursion and new applications, Commun. Number Theory Phys. 9 (2015), 51-187, arXiv:1303.5808.
  10. Borot G., Eynard B., Weisse A., Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces, Selecta Math. (N.S.) 23 (2017), 915-1025, arXiv:1407.4500.
  11. Borot G., Gorin V., Guionnet A., Fluctuations for multi-cut discrete $\beta$-ensembles and application to random tilings, in preparation.
  12. 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.
  13. Borot G., Guionnet A., Asymptotic expansion of $\beta$ matrix models in the multi-cut regime, Forum Math. Sigma 12 (2024), e13, 93 pages, arXiv:1303.1045.
  14. Borot G., Guionnet A., Kozlowski K.K., Large-$N$ asymptotic expansion for mean field models with Coulomb gas interaction, Int. Math. Res. Not. 2015 (2015), 10451-10524, arXiv:1312.6664.
  15. Brézin E., Itzykson C., Parisi G., Zuber J.B., Planar diagrams, Comm. Math. Phys. 59 (1978), 35-51.
  16. Carlson J., Müller-Stach S., Peters C., Period mappings and period domains, Cambridge Stud. Adv. Math., Vol. 168, Cambridge University Press, Cambridge, 2017.
  17. Charlier C., Fahs B., Webb C., Wong M.D., Asymptotics of Hankel determinants with a multi-cut regular potential and Fisher-Hartwig singularities, arXiv:2111.08395.
  18. Claeys T., Grava T., McLaughlin K.D.T.-R., Asymptotics for the partition function in two-cut random matrix models, Comm. Math. Phys. 339 (2015), 513-587, arXiv:1410.7001.
  19. Deift P.A., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lect. Notes Math., Vol. 3, American Mathematical Society, Providence, RI, 1999.
  20. Eynard B., Kimura T., Ribault S., Random matrices, arXiv:1510.04430.
  21. Eynard B., Mehta M.L., Matrices coupled in a chain. I. Eigenvalue correlations, J. Phys. A 31 (1998), 4449-4456, arXiv:cond-mat/9710230.
  22. Farkas H.M., Kra I., Riemann surfaces, Grad. Texts in Math., Vol. 71, 2nd ed., Springer, New York, 1992.
  23. Fay J.D., Theta functions on Riemann surfaces, Lecture Notes in Math., Vol. 352, Springer, Berlin, 1973.
  24. Johansson K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151-204.
  25. Jurkiewicz J., Chaotic behaviour in one-matrix models, Phys. Lett. B 261 (1991), 260-268.
  26. Krichever I.M., Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys 32 (1977), no. 6, 185-213.
  27. Matveev V.B., 30 years of finite-gap integration theory, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), 837-875.
  28. Mehta M.L., Random matrices, 3rd ed., Pure Appl. Math. (Amsterdam), Vol. 142, Elsevier, Amsterdam, 2004.
  29. Migdal A.A., Loop equations and 1/N expansion, Phys. Rep. 102 (1983), 199-290.
  30. Miranda C., Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. 3 (1941), 5-7.
  31. Mulase M., Cohomological structure in soliton equations and Jacobian varieties, J. Differential Geom. 19 (1984), 403-430.
  32. Mumford D., Tata lectures on theta. I, Mod. Birkhäuser Class., Birkhäuser, Boston, MA, 2007.
  33. Pastur L., Limiting laws of linear eigenvalue statistics for Hermitian matrix models, J. Math. Phys. 47 (2006), 103303, 22 pages, arXiv:math.PR/0608719.
  34. Shcherbina M., Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime, J. Stat. Phys. 151 (2013), 1004-1034, arXiv:1205.7062.
  35. Shiota T., Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), 333-382.

Previous article  Next article  Contents of Volume 20 (2024)