Random complex zeroes, III. Decay of the hole probability

By a hole we mean a disc that contains no flat chaotic analytic zero points (i.e. zeroes of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!). A given disc of radius r has a probability of being a hole, - the hole probability. We show that for large r the hole probability decays as exp(-cr^4).