Boutroux curves with external field: equilibrium measures without a minimization problem

Abstract

The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. The applicability of the method ranges from orthogonal polynomials (and generalizations) to Painleve transcendents, and integrable wave equations (KdV, NonLinear Schroedinger, etc.). For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast the requirements for the Cauchy-transform of the equilibrium measure into a problem of algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the ``free boundary problem'', determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees and the support of the measure. The relevance to the quasi--linear Stokes phenomenon for Painleve equations is indicated. A numerical algorithm to find these curves in some cases is also explained. Technical note: the animations included in the file can be viewed using Acrobat Reader 7 or higher. Mac users should also install a QuickTime plugin called Flip4Mac. Linux users can extract the embedded animations and play them with an external program like VLC or MPlayer. All trademarks are owned by the respective companies.

37 pages, 12 figures, 3 animations. Version 2: minor corrections and improved presentation. Version 3: small but critical correction on page 18-19. No change in conclusions

Cited by in corpus (12)

• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Algebraic methods in random matrices and enumerative geometry
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• A holomorphic and background independent partition function for matrix models and topological strings
• Geometry of Spectral Curves and All Order Dispersive Integrable System
• Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve' equation
• Universal scaling limits of matrix models, and (p,q) Liouville gravity
• On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight
• Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights
• Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials
• Valence independent formula for the equilibrium measure
• Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature