Equilibrium measures in the presence of certain rational external fields

TL;DR

This paper studies equilibrium measures on the real axis under rational external fields, analyzing how their support evolves with parameters, with detailed focus on a generalized Gauss-Penner model relevant to random matrix theory.

Contribution

It extends previous work on polynomial external fields to rational external fields, providing detailed analysis and applications to physical models.

Findings

Support of equilibrium measures changes with parameters.

Detailed analysis of the generalized Gauss-Penner model.

Applications to random matrix models.

Abstract

Equilibrium measures in the real axis in the presence of rational external fields are considered. These external fields are called rational since their derivatives are rational functions. We analyze the evolution of the equilibrium measure, and its support, when the size of the measure, $t$, or other parameters in the external field vary. Our analysis is illustrated by studying with detail the case of a generalized Gauss-Penner model, which, in addition to its mathematical relevance, has important physical applications (in the framework of random matrix models). This paper is a natural continuation of \cite{MOR2013}, where equilibrium measures in the presence of polynomial external fields are thoroughly studied.