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.