Transportation to random zeroes by the gradient flow

Details Transportation to random zeroes by the gradient flow Transportation to random zeroes by the gradient flow

2005-10-30

arxiv.org/abs/math/0510654v3

Mathematics

Complex Variables

Improvement of presentation in sections 4 and 11, tiny changes in other sections

Scientific paper

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).

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