Consider the infinite Ginibre ensemble (the distributional limit of the eigenvalues of nxn random matrices with i.i.d. standard complex Gaussian entries) in the complex plane. For a bounded set $U$, let $H_r(U)$ denote the probability (hole probability) that no points of infinite Ginibre ensemble fall in the region $rU$. We study the asymptotic behavior of $H_r(U)$ as $r \to \infty$. Under certain conditions on $U$ we show that $\log H_r(U)=C_U \cdot r^4 (1+o(1))$ as $r \to \infty$. Using potential theory, we give an explicit formula for $C_U$ in terms of the minimum logarithmic energy of the set with a quadratic external field. We calculate $C_U$ explicitly for some special sets such as the annulus, cardioid, ellipse, equilateral triangle and half disk.

Moreover, we generalize the above hole probability results for a class of determinantal point processes in the complex plane.

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Last updated: 17 Jan 2019