Hole probabilities for determinantal point processes in the complex plane

Abstract

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

Ph.D Thesis, 90 pages