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Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. (English) Zbl 1099.60037

From the author’s introduction: Consider the random power series \[ f_U(z)=\sum^\infty_{n=0} a_nz^n,\tag{1} \] where \(\{a_n\}^\infty_{n=0}\) are independent standard complex Gaussian random variables (with density \(e^{-z \overline z}/\pi)\). The radius of convergence of the series is a.s. 1, and the set of zeros forms a point process \(Z_U\) in the unit disk \(U\). Zeros of Gaussian power series have been studied starting with A. C. Offord [Proc. Lond. Math. Soc., III. Ser. 14 A, 199–238 (1965; Zbl 0134.29204)], since these series are limits of random Gaussian polynomials.
Our main new discovery is that the zeros \(Z_U\) form a determinantal process, and this yields an explicit formula for the distribution of the number of zeros in a disk. Furthermore, we show that the process \(Z_U\) admits a conformally invariant evolution which elucidates the repulsion between zeros.

MSC:

60G99 Stochastic processes

Citations:

Zbl 0134.29204
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References:

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