First Colonization of a Spectral Outpost in Random Matrix Theory

We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N−2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature.The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N−γ, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

[1]  Pavel Bleher,et al.  Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model , 1999, math-ph/9907025.

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

[3]  Percy Deift,et al.  New Results on the Equilibrium Measure for Logarithmic Potentials in the Presence of an External Field , 1998 .

[4]  A. Kuijlaars,et al.  Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter , 2002, math/0205175.

[5]  Athanassios S. Fokas,et al.  The isomonodromy approach to matric models in 2D quantum gravity , 1992 .

[6]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[7]  Tom Claeys,et al.  Birth of a Cut in Unitary Random Matrix Ensembles , 2007, 0711.2609.

[8]  Athanassios S. Fokas,et al.  Discrete Painlevé equations and their appearance in quantum gravity , 1991 .

[9]  M. Y. Mo.  The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the , 2007, 0711.3208.

[10]  Athanassios S. Fokas,et al.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: The isomonodromy approach in the theory of two-dimensional quantum gravitation , 1990 .

[11]  Géza Freud.  On polynomial approximation with the weight 363-1363-1363-1 , 1973 .

[12]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[13]  Bertrand Eynard,et al.  Universal distribution of random matrix eigenvalues near the ‘birth of a cut’ transition , 2006, math-ph/0605064.

[14]  Marco Bertola,et al.  Boutroux curves with external field: equilibrium measures without a minimization problem , 2007, 0705.3062.

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

[16]  M. Y. Mo,et al.  Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights , 2006 .

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

[18]  ON POLYNOMIAL APPROXIMATION WITH THE WEIGHT exp{-lx 2~} , 1973 .

[19]  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 .

[20]  Stephanos Venakides,et al.  Strong asymptotics of orthogonal polynomials with respect to exponential weights , 1999 .

[21]  A. S. Fokas,et al.  Matrix models of two-dimensional quantum gravity and isomonodromic solutions of “discrete Painlevé” equations , 1995 .