Boutroux curves with external potential: equilibrium measures without a minimization problem
by
Marco Bertola
Concordia University and CRM

The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast those requirements in a problem in 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. The notion and techniques developed here are not limited to (pseudo) orthogonal polynomials but extend to other settings where the nonlinear steepest descent method is used including Painleve equations. The sudden topological changes (w.r.t. parameters) of the structure of the "free boundary" hinted at earlier are the essence of the so-called "nonlinear Stokes' phenomenon".

Date received: July 4, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavl-03.