The hole probability for Gaussian entire functions

Abstract

Consider the random entire function $$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $$ , where the ϕ n are independent standard complex Gaussian coefficients, and the a n are positive constants, which satisfy $$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $$ . We study the probability P H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a n is logarithmically concave, we prove that $$\log {P_H}(r) = - S(r) + o(S(r))$$ , where $$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $$ , and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.