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

TL;DR

This paper introduces a geometric approach to determine equilibrium measures and their support curves in complex analysis, bypassing traditional minimization problems, with applications to orthogonal polynomials and Painleve equations.

Contribution

It recasts the problem of equilibrium measures into algebraic geometry and harmonic analysis, providing a complete solution without minimization, and addresses the free boundary problem explicitly.

Findings

Solved existence and uniqueness of equilibrium measures without minimization.

Provided a geometric characterization of support curves for orthogonal polynomial zeros.

Developed a numerical algorithm for finding these curves in specific cases.

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.