Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Consider the zero set of the random power series f(z) = P anz n with i.i.d. complex Gaussian coefficientsan. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables Xk, where P(Xk = 1) = r 2k . Moreover, the set of absolute values of the zeros of f has the

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