Equilibrium measures in the presence of certain rational external fields

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 [18], where equilibrium measures in the presence of polynomial external fields are thoroughly studied.

Equilibrium problems in weakly admissible external fields created by pointwise charges

“…In case Q is of the form (3.2) and t < T = 1 − γ, following similar arguments as in [14] and [15], it can be checked that the function R is rational, and more precisely, after taking square root in (3.4), one may prove that…”

“…As in [11] and [14]- [15], we proceed by studying the evolution of the equilibrium measure µ t in the external field Q in (3.2) as the mass t ∈ (0, T ) approaches the limit value T . In the next theorem, we show that the measures µ t have a weak-* limit as t tends to T , namely the equilibrium measure µ T , solution of the weakly admissible equilibrium problem.…”

“…These external fields, or weights, are called rational since their derivatives are rational functions. The particular case where Q has an additional polynomial part of even degree was considered in [14]. Then, in [15], the case of rational external fields of type (1.1) (without polynomial part) was treated when all γ k > 0 (and, thus, all z k ∈ C \ R).…”

“…One of the contributions of this paper is to deal with both positive and negative charges (that is, a mixture of attractors and repellents), where in principle, for γ k < 0, z k ∈ R is allowed. Though, in general, the study of weighted equilibrium problems on a conductor Σ deals with probabilistic equilibrium measures µ Q with µ Q (Σ) = 1, the authors of [11], [14] and [15] have preferred to consider equilibrium measures with varying mass µ t = µ t,Q , in the sense that µ t (Σ) = t > 0. This is similar to the dynamical approach proposed in the seminal work by V. Buyarov and E. A. Rakhmanov [4].…”

“…It is interesting to describe the different possible scenarios for the equilibrium measure in these rational external fields. Also noteworthy is the fact that the weakly admissible external fields considered in this paper may be seen as limits of admissible external fields and, thus, the tools used in the previous papers [11], [14] and [15] for admissible rational external fields will be useful. Finally, the signed equilibrium measure, which is easier to compute than the (positive) equilibrium measure and is closely related to it (see e.g.…”