The Pólya-Tchebotarev problem with semiclassical external fields

TL;DR

The paper addresses a semiclassical extension of the Pólya-Tchebotarev max-min energy problem, seeking a compact, connected contour that contains a prescribed finite set $\\mathcal{C}$ while maximizing the weighted logarithmic energy in an external field $\\varphi=\\mathrm{Re}\,\\Phi$. It develops a rigorous framework based on critical measures and quadratic differentials, extending Rakhmanov–Kuijlaars–Martínez-Finkelshtein methods to external fields with finite singularities, and introducing an admissible contour class encoded by partitions of fixed points and infinity sectors. The main result proves the existence of an extremal contour $\\Gamma_0$ whose equilibrium measure $\\mu_0$ is supported on a finite union of analytic arcs and satisfies a rational spectral curve $R(z)$ given by $R(z)=\bigl(C^{\\mu_0}(z)+\\Phi'(z)\bigr)^2$, with poles tied to the fixed points and poles of $\\Phi$, aligning with the S-property of the zeros distribution in related non-Hermitian orthogonal polynomial problems. The work provides a constructive route to obtain $\\Gamma_0$ from a max-min candidate in an enlarged class and establishes the key geometric and variational properties that connect the extremal measure to contour geometry, thereby unifying several prior PT-type results under semiclassical external fields with finitely many prescribed connections.

Abstract

The classical Pólya-Tchebotarev problem, commonly stated as a max-min logarithmic energy problem, asks for finding a compact of minimal capacity in the complex plane which connects a prescribed collection of fixed points. Variants of this problem have found ramifications and applications in the theory of non-hermitian orthogonal polynomials, random matrices, approximation theory, among others. Here we consider an extension of this classical problem, including a semiclassical external field, and enforcing finitely many prescribed collections of points to be connected, possibly also to infinity. Our method is based on Rakhmanov's approach to max-min problems in logarithmic potential theory, utilizes the developed machinery by Martínez-Finkelshtein and Rakhmanov on critical measures, and extends the development of Kuijlaars and the second named author from the context of polynomial external fields to the semiclassical case considered here.

Figures (9)

• Figure 1: Examples of crossing and non-crossing partitions for a set $\Theta$ of $10$ elements. Any choice of contours that represent the crossing partition have an intersection between at least two contours, and there is no contour representation without this property.
• Figure 2: An example of a set $F$ with the connections between the fixed points.
• Figure 3: An example of a set $F$ with the connections between the fixed points, and connections to infinity through admissible sectors. Note that no matter how you try to connect following the described rules, a connection between $c_1$ and the points $c_2,c_3$ and $c_4$ is established.
• Figure 4: A generic element of $\mathcal{T}$ for the codification given by \ref{['ex:partitions']}.
• Figure 6: A choice for $\gamma$ when $N = 4$. Sets $\Delta_M$ and $\varphi^{-1}(-M)$ are represented by the grey regions and the solid lines, and elements of the sets $\mathcal{C}$ and $\mathcal{Z}$ are represented by black and white dots, respectively. If $\mathcal{T}$ connects $z_1$ to infinity then each contour $\Gamma \in \mathcal{T}$ contain a contour similar to $\Gamma_A$. Analogously, if $\mathcal{T}$ connects infinity through $S_1$ and $S_4$, then $\Gamma$ contains a contour similar to $\Gamma_B$.

Theorems & Definitions (52)

• Example 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Theorem 2.5
• Theorem 2.6: Rakhmanov2012
• Definition 3.1
• Proposition 3.2
• proof
• Definition 3.3