Critical measures for vector energy: Global structure of trajectories of quadratic differentials

Abstract

Abstract. Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector-valued critical measures, a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures µ = (µ 1 , µ 2 , µ 3 ) when the mutual interaction comprises both attracting and repelling forces.For arbitrary vector polynomial external fields we establish general structural results about critical measures, such as their characterization in terms of an algebraic equation solved by an appropriate combination of their Cauchy transforms, and the symmetry properties (or the S-properties) exhibited by such measures. In consequence, we conclude that vector-valued critical measures are supported on a finite number of analytic arcs, that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface. The complete description of the so-called critical graph for such a differential is the key to the construction of the critical measures.We illustrate these connections studying in depth a one-parameter family of critical measures under the action of a cubic external field. This choice is motivated by the asymptotic analysis of a family of (non-hermitian) multiple orthogonal polynomials, that is subject of a forthcoming paper. Here we compute explicitly the Riemann surface and the corresponding quadratic differential, and analyze the dynamics of its critical graph as a function of the parameter, giving a detailed description of the occurring phase transitions. When projected back to the complex plane, this construction gives us the complete family of vector-valued critical measures, that in this context turn out to be vector-valued equilibrium measures.

Supercritical regime for the kissing polynomials

“…To construct the arcs γ 1 and γ 2 , our main tool will be the theory of trajectories of quadratic differentials. We will keep the discussion of the basic theory here to a minimum, and refer to [34,Appendix B] for details. The general theory can be found in the books by Strebel [41] or Pommerenke [37,Chapter 8].…”

Critical measures for vector energy: Asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

“…The present work is a natural continuation of [37]: we analyze a family of non-Hermitian MOPs (both of types I and II) with respect to a cubic weight and find their asymptotic description in terms of the vector critical measures constructed in [37]. It should be pointed out that the large degree zero distribution of these polynomials is highly non-trivial, exhibiting several phase transitions, as Figures 1 and 2 illustrate.…”

“…The structure of the paper is as follows. In the next section we state our main findings that explain in particular the numerical outcomes illustrated in and 2, for which we need to provide a minimum background from [37]. However, the proofs require the use of several technical results from [37] that for convenience of the reader we gather in Sections 3 and 4.…”

“…The interaction matrix considered in [37] appears also in the analysis of random matrix models with external source. For the general quartic potential with even symmetry, and about at the same time, for the general even degree potential with even symmetry, Aptekarev, Lysov and Tulyakov [7,8] and Bleher, Delvaux and Kuijlaars [11], respectively, addressed such matrix model using the MOPs approach.…”

“…Once these trajectories are well understood, the dynamics of the limiting zero distribution can be read off from these trajectories, and the very distinct behaviors portrayed in Figures Figures 1 and 2 are easily understood under the very same framework. Besides the aforementioned quadratic differentials that were previously studied in [37], our main asymptotic tool is the matrix Riemann-Hilbert problem characterization of MOPs [21] with cubic weights and the development of the Deift-Zhou nonlinear steepest descent method that yields not only the limiting zero distribution but a detailed uniform asymptotics of the polynomial on the whole complex plane. In order to be able to perform the necessary steps of this approach we had to introduce a preliminary transformation that although looked purely technical in the first stage, turned out to be crucially related to the core of the matter.…”