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

TL;DR

The paper addresses how to construct a suitable g-function for the nonlinear steepest descent method in rank-two systems with external fields, by recasting equilibrium-measure data as Boutroux curves and harmonic-analytic conditions, thereby avoiding explicit minimization. It develops a complete geometric-analytic framework: Boutroux curves with admissibility, a local coordinate map $\mathcal{P}$ to external data $(V,T)$, and a constructive deformation that preserves a prescribed connectivity pattern, including phase transitions. The main contributions are the existence and (under generic assumptions) uniqueness of simple admissible Boutroux curves compatible with any connectivity pattern for a given $(V,T)$, a reconstruction-from-clock-diagram approach, and a deformation/gluing strategy that yields a global g-function suitable for Deift–Zhou analysis. The work also links these ideas to the quasi-linear Stokes phenomena in Painlevé equations and provides a numerical algorithm for computing Boutroux curves in practice, thereby broadening the applicability to complex-weights orthogonal polynomials and related integrable systems.

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.

Figures (14)

• Figure 1: The surface of (the arctan of) $h(x)$ (see the text for explanation) for an admissible Boutroux simple curve with external potential $V(x) = x^6/6$. Note the "creases" where $h$ is clearly non-differentiable but continuous; on each side of each crease the surface is negative.
• Figure 2: An example of contours for a potential $V(x)$ of degree $8$. The shaded sectors are "forbidden" directions of approach at infinity. The contours $\gamma_\ell$ can approach $\infty$ along any direction in the "allowed sectors", non shaded and marked as $\mathcal{S}_1, \dots, \mathcal{S}_{d+1}$. We should think of the oriented contours $\gamma_\ell$ as a wire or highway carrying a current (traffic) $\varkappa_\ell\in {\mathbb C}$: the total incoming current in a sector $\mathcal{S}_j$ is the sum of the currents in/out carried by all contours accessing that particular sector. Note that the total traffic in/out of all sectors is zero.
• Figure 3: A pictorial representation of the surface of the discriminant $\bf\Sigma$ in the $(V,T)$--space: it has singularities corresponding to the subloci where the Boutroux curves become more and more degenerate as well as self-intersections. Our path is chosen to intersect it in the "smooth" part corresponding to the simple but (simply) critical Boutroux curves and transversally (i.e. the tangent to the path is not tangent to the discriminant). In the picture, the arc represents a path in the $\mathcal{V}$ space, and indicated is the tangent vector at the crossing of the discriminant. Note that the picutre is three--dimensional, but actual situation are always of higher dimension, so that this is only a suggestive picture.
• Figure 4: Using the same example of Fig. \ref{['contours']} with some specific values of the currents $\varkappa_\ell$ (indicated near the tip of the arrows). The sectors with zero total incoming current can be skipped by the irreducible connectivity pattern and the contours carrying zero current are irrelevant. In this example there are two notably distinct irreducible connectivity patterns because the highways carry the same total traffic.
• Figure 5: The construction of the causeways/oceans from the irreducible connectivity pattern. The two clock diagrams correspond to the two equivalent irreducible patterns in Fig. \ref{['totalcurrent']}. Since two highways have the same traffic (see Fig. \ref{['totalcurrent']}) we have the two possibilities above. The two cases correspond to an exchange re-routing.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.1
• Definition 2.5
• Proposition 2.1
• Remark 2.2
• Lemma 2.1
• Definition 2.6