Boutroux curves with external field: equilibrium measures without a variational 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 Painlevé transcendents, and integrable wave equations (KdV, NonLinear Schrödinger, 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 Painlevé equations is indicated. A numerical algorithm to find these curves in some cases is also explained.

Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence

“…In fact, for the 1-matrix model with polynomial potential, this can be proved a posteriori from the asymptotics of M. Bertola [6,7]. But for more general cases, it is only a conjecture, for instance for the 2-matrix model.…”

“…From [6], it should be expected that if we choose ǫ * such that the spectral curve has the Boutroux property:…”

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

“…The theory quickly developed and found applications into such fields as the Riemann-Hilbert approach to strong asymptotics, random matrix theory [10,4,5,6,3,2] and in the study of dualities between supersymmetric gauge theories and string models [18,7,11,12,15]. This paper is devoted to a detailed analysis of the phase structure and phase transitions of the asymptotic (in the limit n → ∞) zero density of monic orthogonal polynomials P n (z) = z n + · · · Γ P n (z)z k e −nW (z) dz = 0, k = 0, .…”

Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves

“…It was then used in [5] to construct a "g-function", an integral part of the Deift-Zhou [6] steepest descent analysis of the asymptotics of non-hermitean complex orthogonal polynomials. In an analogous context it appears implicitly in the study of the semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation [7].…”

“…For simple degenerations (nodes) of a hyperelliptic surface R the problem has been, in fact, addressed in [5,7] (in different language and with different intents). For general compact Riemann surfaces this same problem was studied in [3].…”