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

By Yuval Peres, B'alint Vir'ag. 2003

180 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7099761d56368de3cc31d924b3c8b428e54e036a

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Citations in the corpus

Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, methodology): “ee the recent text [29] and the references therein). In that case, the k-point correlation functions are given by the determinantal formula ˆ(k) GPS (z 1;:::;z k) = 1 ˇk det( 1 1 z i j ) 1i;jk; see [36]. One may then hope that a somewhat analogous formula might be obtained for the random Laurent ser”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, background): “From Peres and Virág [42], it is known that this level set forms a determinantal point process.”
The Noise in the Circular Law and the Gaussian Free Field (2006, background): “This observation may be gleaned from Hammersely [10], but see also [17] for why the adjective “truncated” is used.”
The Palm measure and the Voronoi tessellation for the Ginibre process (2006, unknown): “Similar results could be proved for the point process in the unit disk of C related to the Bergman kernel and studied in [24].”
Around the circular law (2011, background): “The complex Gaussian case was considered by Leboeuf [98] and by Peres and Virág [118], while the real roots of the real Gaussian case were studied by Schehr and Majumdar [131].”
Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions (2005, methodology): “as discovered by Peres and Vir´ag [ 6 ].”
Determinantal Processes and Independence (2005, background): “Now apply the classical Wick formula (see [11] or [23], §2, Fact 10) to get the result.”
Zeros of Random Analytic Functions (2006, methodology): “By the result of Peres and Virág quoted in Theorem 4.1.2, the set of absolute values of the zeros of f1 has the same distribution as the set {U1/2nn } where Un are i.i.d. uniform[0, 1] random variables.”