Hole probabilities for determinantal point processes in the complex plane

We study the hole probabilities for ${\mathcal{X}}_{\mathrm{\infty }}^{(\alpha )}$ ($\alpha >0$), 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 ${\mathcal X}{\infty}^{(\alpha)}(rU)$denotethenumberofpointsof${\mathcal X}{\infty}^{(\alpha)}$thatfallin$rU$\mathrm{.}Then,undersomeconditionson$U$, we show that $$\underset{r\to \mathrm{\infty }}{\lim }\frac{1}{r^{2\alpha }}\log \mathbb{P}[{\mathcal{X}}_{\mathrm{\infty }}^{(\alpha )}(rU)=0]=R_{\mathrm{\varnothing }}^{(\alpha )}-R_U^{(\alpha )},$$ where $\mathrm{\varnothing }$ is the empty set and $$R_U^{(\alpha )}:=\underset{\mu \in \mathcal{P}(U^c)}{\inf }\{\iint \log \frac{1}{\mathrm{\mid }z-w\mathrm{\mid }}d\mu (z)d\mu (w)+\int \mathrm{\mid }z{\mathrm{\mid }}^{\alpha }d\mu (z)\},$$ $\mathcal{P}(U^c)$ 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 $\frac{\mathrm{\mid }z{\mathrm{\mid }}^{\alpha }}{2}$. 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.