论文标题
复杂随机矩阵模型的常规制度的开放性
Openness of Regular Regimes of Complex Random Matrix Models
论文作者
论文摘要
考虑一般复杂的多项式外部字段$$ V（z）= \ frac {z^{k}} {k}+\ sum_ {j = 1}^{k-1}^{k-1} \ frac {t_j z^j^j} { $$修复了可允许的轮廓的等价类$ \ Mathcal {t} $，其成员在两个不同的方向上接近$ \ infty $，并考虑关联的最大能量问题。 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} $。我们使用隐式函数定理来证明端点方程可在一个常规$ q $ cut点的足够小社区中解决。我们还建立了所有$ q $ -cut制度的终点的真实和虚构部分的实用分析，$ 1 \ leq q \ leq 2p-1 $，就外部字段中复杂参数的真实和虚构部分而言。我们选择的$ k $和等价类$ \ m natercal {t} \ ni \ ni \ mathbb {r} $的可允许轮廓仅仅是为了简单，我们的证明以类似的方式扩展到所有可能的选择。
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.
