Transportation to Random Zeroes by the Gradient Flow

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 $e^{-cR^{4}(logR)^{-3/2}}$ .