Openness of Regular Regimes of Complex Random Matrix Models

Abstract

Fix an equivalence class T of admissible contours whose members approach ∞ in two different directions and consider the associated max-min energy problem [14]. When 𝑘 = 2 𝑝, 𝑝 ∈ N, and T contains the real axis, we show that the set of parametersWe 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 ≤ 𝑞 ≤ 2 𝑝 − 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 T R of admissible contours is only for the simplicity of exposition and our proof extends to all possible choices in an analogous way.