Transportation to random zeroes by the gradient flow

2005/10/30 by Nazarov, Fedor, Sodin, Mikhail, Volberg, Alexander · 3 citations
#30B20 #30C15 #60G60 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0510654

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(-R⁴). 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(-cR⁴).

Cited by

• Local repulsion between zeros and critical points of the Gaussian Entire Function
• Sampling properties of the zeroes of the Gaussian entire function
• Optimal matchings of randomly perturbed lattices

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