Abstract
We investigate the matrix model with weight
and unitary symmetry. In particular we study the double scaling limit as \({N \to \infty}\) and \({(\sqrt{N} t, Nz^2 ) \to (u_1,u_2)}\), where N is the matrix dimension and the parameters (u 1, u 2) remain finite. Using the Deift-Zhou steepest descent method, we compute the asymptotics of the partition function when z and t are of order \({O\bigl(N^{-1/2} \bigr)}\). In this regime we discover a phase transition in the (z, N)-plane characterised by the Painlevé III equation. This is the first time that Painlevé III appears in studies of double scaling limits in Random Matrix Theory and is associated to the emergence of an essential singularity in the weighting function. The asymptotics of the partition function is expressed in terms of a particular solution of the Painlevé III equation. We derive explicitly the initial conditions in the limit \({Nz^2\rightarrow u_2}\) of this solution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams M.R., Harnad J., Hurtubise J.: Isospectral Hamiltonian flows in finite and infinite dimensions. II. Integration of flows. Commun. Math. Phys. 134(3), 555–585 (1990)
Adams M.R., Harnad J., Hurtubise J.: Darboux coordinates and Liouville-Arnold integration in loop algebra. Commun. Math. Phys. 155(2), 385–413 (1993)
Adams M.R., Harnad J., Previato E.: Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loops algebra. Commun. Math. Phys. 117(3), 451–500 (1988)
Berry M.V., Shukla P.: Tuck’s incompressibility function: statistics for zeta zeros and eigenvalues. J. Phys. A Math. Theor. 41(38), 385202 (2008)
Bertola M., Eynard B., Harnad J.: Semiclassical orthgonal polynomials, matrix models, and isomonodromic tau functions. Commun. Math. Phys. 263(2), 401–437 (2006)
Bertola, M., Harnad, J., Hurtubise, J., Pusztai, G.: Private communication (2004)
Bleher P., Its A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185–266 (1999)
Bleher P., Its A.: Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Commun. Pure Appl. Math. 56(4), 433–516 (2003)
Brouwer P.W., Frahm K.M., Beenakker C.W.J.: Quantum mechanical time-delay matrix in chaotic scattering. Phys. Rev. Lett. 78(25), 4737–4740 (1997)
Chen Y., Its A.: Painlevé III and a singular linear statistics in Hermitian random matrix ensembles. I. J. Approx. Theory 162(2), 270–297 (2010)
Chen, Y., Its, A.: Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, II. The Asymptotic analysis, unpublished (2009)
Claeys T., Kuijlaars BJ: Universality of the double scaling limit in random matrix models. Commun. Pure Appl. Math. 59(11), 1573–1603 (2006)
Claeys T., Kuijlaars B.J., Vanlessen M.: Multi-critical unitary random matrix ensembles and the general Painlevé II equation. Ann. Math. 168(2), 601–642 (2008)
Claeys T., Vanlessen M.: Universality of a double scaling limit near singular edge points in random matrix models. Commun. Math. Phys. 273(2), 499–532 (2007)
Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. In: Courant Lecture Notes in Mathematics, vol. 3. New York University Courant Institute of Mathematical Sciences, New York (1999)
Dueñez E., Farmer D.W., Froehlich S., Hughes C.P., Mezzadri F., Phan T.: Roots of the derivative of the Riemann-zeta function and of characteristic polynomials. Nonlinearity 23(10), 2599–2621 (2010)
Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999)
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. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999)
Deift P., Zhou X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137(2), 295–368 (1993)
Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons. In: Springer Series in Soviet Mathematics. Springer, Berlin (1987)
Fokas A.S., Its A.R., Kitaev A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142(2), 313–344 (1991)
Fokas A.S., Its A.R., Kitaev A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992)
Forrester P.J., Witte N.S.: Application of the τ-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55(6), 679–727 (2002)
Forrester P.J., Witte N.S.: Boundary conditions associated with the Painlevé III’ and V evaluations of some random matrix averages. J. Phys. A Math. Gen. 39(28), 8983–8995 (2006)
Harnad J.: Dual isomonodromic deformations and moments maps to loop algebras. Commun. Math. Phys. 166(2), 337–365 (1994)
Harnad, J.: The Haminltonian structure of the general rational isomonodromic deformations. Talk at the Colloque international en l’ honneur de Pierre van Moerbeke, Poitiers, France (2005). Unpublished
Harnad J., Routhier M.: R-matrix construction of electromagnetic models for the Painlevé transcendents. J. Math. Phys. 36(9), 4863–4881 (1995)
Jimbo M., Miwa T., Ueno K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function. Phys. D 2(2), 306–352 (1981)
Lukyanov S.: Finite temperature expectation values of local fields in the sinh-Gordon model. Nucl. Phys. B 612(3), 391–412 (2001)
Mezzadri, F.: Random matrix theory and the zeros of ζ′(s). J. Phys. A Math. Gen. 36(12), 2945–2962 (2003) (Random matrix theory)
Mazzocco M., Mo M.Y.: The Hamiltonian structure of the second Painlevé hierarchy. Nonlinearity 20(12), 2845–2882 (2007)
Mezzadri F., Mo M.Y.: On an average over the Gaussian Unitary Ensemble. Int. Math. Res. Not. 2009(18), 3486–3515 (2009)
Mezzadri F., Simm N.J.: Tau-function theory of quantum chaotic transport with β = 1,2,4. Commun. Math. Phys. 324(2), 465–513 (2013)
Szegő, G.: Orthogonal Polynomials. Colloquium Publications, vol. 23. American Mathematical Society, New York (1939)
Texier C., Majumdar S.N.: Wigner time-delay distribution in chaotic cavities and freezing transition. Phys. Rev. Lett. 110(25), 250602 (2013)
Veselov, A.P., Novikov, S.P.: Poisson brackets and complex tori, Algebraic geometry and its applications. Trudy Mat. Inst. Steklov. vol. 165, pp. 49–61. MAIK Nauka/Interperiodica (1984)
Zhou X.: The Riemann-Hilbert problem and inverse scattering. SIAM J. Math. Anal. 20(4), 966–986 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Deift
The authors acknowledge financial support by the EPSRC grant EP/G019843/1.
Rights and permissions
About this article
Cite this article
Brightmore, L., Mezzadri, F. & Mo, M.Y. A Matrix Model with a Singular Weight and Painlevé III. Commun. Math. Phys. 333, 1317–1364 (2015). https://doi.org/10.1007/s00220-014-2076-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2076-z