Hole probabilities for finite and infinite Ginibre ensembles

TL;DR

This paper investigates the probabilities of finding no points in certain regions for the infinite Ginibre ensemble, using potential theory to derive explicit formulas for these probabilities in various geometric settings.

Contribution

It provides explicit formulas for hole probabilities in the infinite Ginibre ensemble using potential theory, including calculations for specific geometric regions.

Findings

Derived explicit formulas for hole probabilities in the Ginibre ensemble.

Calculated minimum energy measures for various geometric sets.

Connected hole probabilities to potential theory and energy minimization.

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 $$ \lim_{r\to \infty}\frac{1}{r^4}\log\mathbb P[\mathcal X_{\infty}(rU)=0]=R_{\emptyset}-R_{U}, $$ where $\emptyset$ is the empty set and $$ R_U:=\inf_{\mu\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}d\mu(z)d\mu(w)+\int |z|^2d\mu(z) \right\}, $$ $\mathcal P(U^c)$ 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.