The P\'olya-Tchebotarev problem with semiclassical external fields

TL;DR

This paper extends the classical Pólya-Tchebotarev problem by incorporating semiclassical external fields and multiple connection constraints, using advanced potential theory methods to analyze minimal capacity configurations.

Contribution

It introduces a novel extension of the classical problem with semiclassical external fields and multiple connection points, expanding the theoretical framework and solution techniques.

Findings

Extended the Pólya-Tchebotarev problem to include semiclassical external fields.

Developed new methods based on Rakhmanov's approach and critical measures.

Provided insights applicable to orthogonal polynomials, random matrices, and approximation theory.

Abstract

The classical P\'olya-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 Mart\'inez-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.