Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

We apply the method we have described in a previous paper (2013) to determine the phase structure of asymptotic zero densities of the standard cubic model of non-Hermitian orthogonal polynomials. We provide a complete description of the two phases: the one cut phase and the two cut phase, and analyze the phase transition processes of the types: splitting of a cut, birth and death of a cut.

[1]  E. Rakhmanov,et al.  EQUILIBRIUM MEASURE AND THE DISTRIBUTION OF ZEROS OF EXTREMAL POLYNOMIALS , 1986 .

[2]  A large N duality via a geometric transition , 2001, hep-th/0103067.

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

[4]  Phase structure of a brane/anti-brane system at large N , 2007, hep-th/0702077.

[5]  C. Vafa,et al.  On geometry and matrix models , 2002, hep-th/0207106.

[6]  Daan Huybrechs,et al.  Zero distribution of complex orthogonal polynomials with respect to exponential weights , 2013, J. Approx. Theory.

[7]  P. Bleher.  Lectures on random matrix models. The Riemann-Hilbert approach , 2008, 0801.1858.

[8]  Marco Bertola,et al.  Boutroux curves with external field: equilibrium measures without a variational problem , 2011 .

[9]  E. Rakhmanov,et al.  Orthogonal Polynomials and $S$-curves , 2011, 1112.5713.

[10]  Herbert Stahl,et al.  Extremal domains associated with an analytic function I , 1985 .

[11]  M. Mariño,et al.  Large N duality beyond the genus expansion , 2009, 0911.4692.

[12]  Daan Huybrechs,et al.  Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature , 2010, J. Approx. Theory.

[13]  Pavel Bleher,et al.  Double scaling limit in the random matrix model: The Riemann‐Hilbert approach , 2002, math-ph/0201003.

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

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

[16]  Matrix Models , 2005, hep-th/0506186.

[17]  Herbert Stahl,et al.  Orthogonal polynomials with complex-valued weight function, I , 1986 .

[18]  Elena Medina,et al.  Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials , 2013, 1305.3028.

[19]  Herbert Stahl,et al.  Orthogonal polynomials with complex-valued weight function, II , 1986 .

[20]  Arno B. J. Kuijlaars,et al.  S-curves in polynomial external fields , 2013, J. Approx. Theory.

[21]  F. David.  Phases of the large-N matrix model and non-perturbative effects in 2D gravity , 1991 .

[22]  渋谷 泰隆.  Global theory of a second order linear ordinary differential equation with a polynomial coefficient , 1975 .

[23]  E. Rakhmanov,et al.  EQUILIBRIUM DISTRIBUTIONS AND DEGREE OF RATIONAL APPROXIMATION OF ANALYTIC FUNCTIONS , 1989 .

[24]  C. Vafa,et al.  Matrix models, topological strings, and supersymmetric gauge theories , 2002, hep-th/0206255.

[25]  M. van der Ploeg,et al.  Matrix models , 2004, Histochemistry.

[26]  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 .

[27]  E. Rakhmanov,et al.  Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials , 2009, 0902.0193.