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

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.