Hole probabilities for finite and infinite Ginibre ensembles

Abstract

We study the hole probabilities of the infinite Ginibre ensemble ${\mathcal{X}}_{\infty }$, a determinantal point process on the complex plane with the kernel $\mathbb{K}(z,w)=\frac{1}{\pi }{e}^{z\bar{w}-\frac{1}{2}|z{|}^2-\frac{1}{2}|w{|}^2}$ with respect to the Lebesgue measure on the complex plane. Let $U$ be an open subset of open unit disk $\mathbb{D}$ and ${\mathcal{X}}_{\infty }(rU)$ denote the number of points of ${\mathcal{X}}_{\infty }$ that fall in $rU$. Then, under some conditions on $U$, we show that $$\underset{r\to \infty }{\lim }\frac{1}{r^4}\log \mathbb{P}[{\mathcal{X}}_{\infty }(rU)=0]=R_{\varnothing }-R_U,$$ where $\varnothing$ is the empty set and $$R_U:=\underset{\mu \in \mathcal{P}(U^c)}{\inf }\left\{\iint \log \frac{1}{|z-w|}d\mu (z)d\mu (w)+\int |z{|}^{2}d\mu (z)\right\},$$ $\mathcal{P}(U^c)$ is the space of all compactly supported probability measures with support in $U^c$. Using potential theory, we give an explicit formula for $R_U$, the minimum possible energy of a probability measure compactly supported on $U^c$ under logarithmic potential with a quadratic external field. Moreover, we calculate $R_U$ explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.

28 pages. To appear in International Mathematics Research Notices (IMRN)