Hole probabilities for determinantal point processes in the complex plane
We study the hole probabilities for ${\mathcal X}_{\infty}^{(α)}$ ($α>0$), a determinantal point process in the complex plane with the kernel $\mathbb K_{\infty}^{(α)}(z,w)=\fracα{2π}E_{\frac{2}α,\frac{2}α}(z\bar w)e^{-\frac{|z|^α}{2}-\frac{|w|^α}{2}}$ 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 $D(0,(\frac{2}α)^{\frac{1}α})$ and ${\mathcal X}_{\infty}^{(α)}(rU)$ denote the number of points of ${\mathcal X}_{\infty}^{(α)}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \lim_{r\to \infty}\frac{1}{r^{2α}}\log\mathbb P[\mathcal X_{\infty}^{(α)}(rU)=0]=R_{\emptyset}^{(α)}-R_{U}^{(α)}, $$ where $\emptyset$ is the empty set and $$ R_U^{(α)}:=\inf_{μ\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}dμ(z)dμ(w)+\int |z|^αdμ(z) \right\}, $$ $\mathcal P(U^c)$ is the space of all compactly supported probability measures with support in $U^c$. Using potential theory, we give an explicit formula for $R_U^{(α)}$, the minimum possible energy of a probability measure compactly supported on $U^c$ under logarithmic potential with an external field $\frac{|z|^α}{2}$. In particular, $α=2$ 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.