Abstract.
We consider the zeroes of the random Gaussian entire function \(\mathop {f(z)=\sum\limits^{\infty}_{k=0}} \xi k \frac{z^{k}}{\sqrt{k!}}\) (\(\xi_{0}, \xi_{1}\) , . . . are Gaussian i.i.d. complex random variables) and show that their basins under the gradient flow of the random potential \(U(z) = log |f(z)| - \frac{1}{2}|z|^{2}\) partition the complex plane into domains of equal area.
We find three characteristic exponents 1, 8/5, and 4 of this random partition: the probability that the diameter of a particular basin is greater than R is exponentially small in R; the probability that a given point z lies at a distance larger than R from the zero, it is attracted to decays as \(e^{-R^{8/5}}\) ; and the probability that, after throwing away 1% of the area of the basin, its diameter is still larger than R decays as \(e^{-R^{4}}\) .
We also introduce a combinatorial procedure that modifies a small portion of each basin in such a way that the probability that the diameter of a particular modified basin is greater than R decays as \(e^{-cR^{4}(logR)^{-3/2}}\) .
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F.N. and A.V. partially supported by the National Science Foundation, DMS grant 0501067. M.S. partially supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 357/04.
Received: May 2006 Revision: February 2007 Accepted: March 2007
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Nazarov, F., Sodin, M. & Volberg, A. Transportation to Random Zeroes by the Gradient Flow. GAFA, Geom. funct. anal. 17, 887–935 (2007). https://doi.org/10.1007/s00039-007-0613-z
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DOI: https://doi.org/10.1007/s00039-007-0613-z