Asymptotics for a special solution of the thirty fourth Painlevé equation

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z(n,N)(-1) vertical bar det M vertical bar(2 alpha)e(-NTrV(M)) dM with alpha > -1/2. The factor vertical bar det M vertical bar(2 alpha) induces critical eigenvalue behaviour near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N - 1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention was on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution u(alpha) of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which, however, is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behaviour of u(alpha)(s) as s -> +/-infinity. We conjecture that this asymptotics characterizes u(alpha) and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u(alpha). We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.

[1]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[2]  A. Newell,et al.  Monodromy- and spectrum-preserving deformations I , 1980 .

[3]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III , 1981 .

[4]  Michio Jimbo,et al.  Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function , 1981 .

[5]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[6]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[7]  A. Kapaev.  Global asymptotics of the second Painlevé transcendent , 1992 .

[8]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[9]  C. Tracy,et al.  Level-spacing distributions and the Airy kernel , 1992, hep-th/9211141.

[10]  Airy Kernel and Painleve II , 1999, solv-int/9901004.

[11]  E. Hubert,et al.  A note on the Lax pairs for Painlevéequations , 1999 .

[12]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[13]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[14]  C. Howls,et al.  Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear , 2000 .

[15]  Alexander Its,et al.  Isomonodromic Deformations and Applications in Physics , 2002 .

[16]  A. Kuijlaars,et al.  Universality for Eigenvalue Correlations at the Origin of the Spectrum , 2003, math-ph/0305044.

[17]  A. Its,et al.  Quasi-linear Stokes phenomenon for the second Painlevé transcendent , 2001, nlin/0108010.

[18]  Maarten Vanlessen,et al.  Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight , 2002, J. Approx. Theory.

[19]  A. Kapaev.  Quasi-linear Stokes phenomenon for the Hastings-McLeod solution of the second Painlevé equation , 2004, nlin/0411009.

[20]  D. Shih,et al.  Flux vacua and branes of the minimal superstring , 2004, hep-th/0412315.

[21]  A. Kuijlaars,et al.  Multi-critical unitary random matrix ensembles and the general Painlevé II equation , 2005 .

[22]  Athanassios S. Fokas,et al.  Painleve Transcendents: The Riemann-hilbert Approach , 2006 .

[23]  A. Kuijlaars,et al.  Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painlevé Transcendent , 2007, 0704.1972.

[24]  A. Its,et al.  Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump , 2007, 0706.3192.

[25]  P. Deift,et al.  Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities , 2009, 0905.0443.