Openness of Regular Regimes of Complex Random Matrix Models

TL;DR

This paper proves that the set of parameters leading to regular multi-cut equilibrium measures in complex random matrix models is open and that the endpoints of these measures depend real-analytically on the parameters, using the implicit function theorem.

Contribution

It establishes the openness of the parameter set for regular q-cut measures and proves real-analytic dependence of endpoints on parameters in complex polynomial external fields.

Findings

The set of parameters producing regular q-cut measures is open in complex space.

Endpoints of the measures depend real-analytically on external field parameters.

The methods extend to all admissible contours beyond the real axis.

Abstract

Consider the general complex polynomial external field $$ V(z)=\frac{z^{k}}{k}+\sum_{j=1}^{k-1} \frac{t_j z^j}{j}, \qquad t_j \in \mathbb{C}, \quad k \in \mathbb{N}. $$ Fix an equivalence class $\mathcal{T}$ of admissible contours whose members approach $\infty$ in two different directions and consider the associated max-min energy problem. When $k=2p$, $p \in \mathbb{N}$, and $\mathcal{T}$ contains the real axis, we show that the set of parameters $t_1, \cdots, t_{2p-1}$ which gives rise to a regular $q$-cut max-min (equilibrium) measure, $1 \leq q \leq 2p-1 $, is an open set in $\mathbb{C}^{2p-1}$. We use the implicit function theorem to prove that the endpoint equations are solvable in a small enough neighborhood of a regular $q$-cut point. We also establish the real-analyticity of the real and imaginary parts of the end-points for all $q$-cut regimes, $1 \leq q \leq 2p-1$, with respect to the real and imaginary parts of the complex parameters in the external field. Our choice of even $k$ and the equivalence class $\mathcal{T} \ni \mathbb{R}$ of admissible contours is only for the simplicity of exposition and our proof extends to all possible choices in an analogous way.