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–>\infty. Under certain conditions on U we show that \log H_r(U)=C_U.r^4 (1+o(1)) as r–> \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.