O ct 2 00 3 Zeros of the i . i . d . Gaussian power series : a conformally invariant determinantal process

Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. 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 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) = r2k. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.