### Random Matrices with Merging Singularities and the Painlevé V Equation

Tom Claeys and Benjamin Fahs
Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-La-Neuve, Belgium

Received September 08, 2015, in final form March 18, 2016; Published online March 23, 2016

Abstract
We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $\alpha>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of $\alpha$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

Key words: random matrices; Painlevé equations; Riemann-Hilbert problems.

pdf (724 kb)   tex (45 kb)

References

1. Akemann G., Dalmazi D., Damgaard P.H., Verbaarschot J.J.M., ${\rm QCD}_3$ and the replica method, Nuclear Phys. B 601 (2001), 77-124, hep-th/0011072.
2. Atkin M.R., A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential, arXiv:1504.04539.
3. Brézin E., Hikami S., Characteristic polynomials of random matrices, Comm. Math. Phys. 214 (2000), 111-135, math-ph/9910005.
4. Claeys T., Its A., Krasovsky I., Emergence of a singularity for Toeplitz determinants and Painlevé V, Duke Math. J. 160 (2011), 207-262, arXiv:1004.3696.
5. Claeys T., Krasovsky I., Toeplitz determinants with merging singularities, Duke Math. J. 164 (2015), 2897-2987, arXiv:1403.3639.
6. Damgaard P.H., Nishigaki S.M., Universal massive spectral correlators and three-dimensional QCD, Phys. Rev. D 57 (1998), 5299-5302, hep-th/9711096.
7. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 1999.
8. Deift P., Its A., Krasovsky I., Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math. 174 (2011), 1243-1299, arXiv:0905.0443.
9. Deift P., Its A., Krasovsky I., Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results, Comm. Pure Appl. Math. 66 (2013), 1360-1438, arXiv:1207.4990.
10. Deift P., Kriecherbauer T., McLaughlin K.T.-R., New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388-475.
11. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999), 1491-1552.
12. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335-1425.
13. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368, math.AP/9201261.
14. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: the Riemann-Hilbert approach, Mathematical Surveys and Monographs, Vol. 128, Amer. Math. Soc., Providence, RI, 2006.
15. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in $2$D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
16. Forrester P.J., Frankel N.E., Applications and generalizations of Fisher-Hartwig asymptotics, J. Math. Phys. 45 (2004), 2003-2028, math-ph/0401011.
17. Forrester P.J., Frankel N.E., Garoni T.M., Witte N.S., Finite one-dimensional impenetrable Bose systems: occupation numbers, Phys. Rev. A 67 (2003), 043607, 17 pages, cond-mat/0211126.
18. Forrester P.J., Witte N.S., Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\rm P}_{\rm VI}$, the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J. 174 (2004), 29-114, math-ph/0204008.
19. Fyodorov Y.V., Khoruzhenko B.A., Simm N.J., Fractional Brownian motion with Hurst index $H=0$ and the Gaussian unitary ensemble, Ann. Probab., to appear, arXiv:1312.0212.
20. Fyodorov Y.V., Simm N.J., On the distribution of the maximum value of the characteristic polynomial of GUE random matrices, arXiv:1503.07110.
21. Garoni T.M., On the asymptotics of some large Hankel determinants generated by Fisher-Hartwig symbols defined on the real line, J. Math. Phys. 46 (2005), 043516, 19 pages, math-ph/0411019.
22. Its A.R., Kuijlaars A.B.J., Östensson J., Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent, Int. Math. Res. Not. 2008 (2008), no. 9, rnn017, 67 pages, arXiv:0704.1972.
23. Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
24. Jimbo M., Miwa T., Studies on holonomic quantum fields. XVII, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 405-410.
25. Krasovsky I., Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant, Duke Math. J. 139 (2007), 581-619, math-ph/0411016.
26. Kuijlaars A.B.J., McLaughlin K.T.-R., Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), 736-785.
27. Kuijlaars A.B.J., Vanlessen M., Universality for eigenvalue correlations at the origin of the spectrum, Comm. Math. Phys. 243 (2003), 163-191, math-ph/0305044.
28. Mu\ugan U., Fokas A.S., Schlesinger transformations of Painlevé ${\rm II}$-${\rm V}$, J. Math. Phys. 33 (1992), 2031-2045.
29. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W., NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010.
30. Saff E.B., Totik V., Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Berlin, 1997.
31. Vanlessen M., Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, J. Approx. Theory 125 (2003), 198-237, math.CA/0212014.
32. Wu T.T., McCoy B.M., Tracy C.A., Barouch E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B 13 (1976), 316-374.
33. Xu S.-X., Zhao Y.Q., Critical edge behavior in the modified Jacobi ensemble and Painlevé equations, Nonlinearity 28 (2015), 1633-1674, arXiv:1404.5105.