Transportation to random zeroes by the gradient flow

Abstract

We consider the zeroes of a random Gaussian Entire Function f and show that their basins under the gradient flow of the random potential U 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 exp(-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 exp(-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 only slightly slower than exp(-cR^4).