Hole probabilities for determinantal point processes in the complex plane

TL;DR

This paper investigates the probabilities of finding no points in certain regions for a class of determinantal point processes in the complex plane, providing explicit formulas and calculations for specific shapes.

Contribution

It derives explicit formulas for hole probabilities of determinantal point processes with Mittag-Leffler kernels, extending potential theory methods to complex geometric regions.

Findings

Explicit formula for hole probabilities involving energy minimization

Calculation of energy for specific geometric regions

Connection to the infinite Ginibre ensemble for alpha=2

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 $$ \lim_{r\to \infty}\frac{1}{r^{2\alpha}}\log\mathbb P[\mathcal X_{\infty}^{(\alpha)}(rU)=0]=R_{\emptyset}^{(\alpha)}-R_{U}^{(\alpha)}, $$ where $\emptyset$ is the empty set and $$ R_U^{(\alpha)}:=\inf_{\mu\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}d\mu(z)d\mu(w)+\int |z|^{\alpha}d\mu(z) \right\}, $$ $\mathcal P(U^c)$ 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.