论文标题
Berezin密度和平面正交多项式
Berezin density and planar orthogonal polynomials
论文作者
论文摘要
我们为Laplacian介绍了一个非线性潜在理论问题，该解决方案是为多项式Bergman空间的Berezin密度$ B（Z，\ CDOT）$的特征，其中固定了点$ z \ in \ mathbb {c} $。当$ z = \ infty $时，用相应的归一化正交多项式$ p $的平方模量表示berezin密度。我们使用此表征的大概版本来研究在指数变化的权重的背景下正交多项式的渐近性。这是建立在Takhtajan的早期作品的基础上的，第一作者是针对平面正交多项式的柔和的Riemann-Hilbert问题，在那里，我们拥有$ \ bar \ bar \ partial $ operator。 我们适应了柔软的Riemann-Hilbert方法，以解决非线性潜在问题，在非线性潜在问题中，非线性是由于出现$ | p |^2 $，而不是$ \ overline {p} $。此外，我们建议如何适应潜在的理论方法，以研究更一般的berezin密度$ b（z，w）$在裸露政权中，即当$ z $固定在液滴外时。这是程序中的第一部分，可为多项式Bergman内核获得明确的全局扩展公式，尤其是相关的随机正常矩阵集合的单点函数。
We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial $P$. We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the $\bar\partial$-operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of $|P|^2$ in place of $\overline{P}$. Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities $B(z,w)$ in the off-spectral regime, that is, when $z$ is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.
