  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.
