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

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

2003-10-19

arxiv.org/abs/math/0310297v4

Acta Math. 194 (2005), no. 1, 1-35

Mathematics

Probability

37 pages, 2 figures, updated proofs

Scientific paper

10.1007/BF02392515

Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. 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 X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. 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.

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