2019/02/22 by Zakhar Kabluchko, Kabluchko, Zakhar, Igor Wigman +1
Earth and Planetary Sciences · Mathematics · #14P05 #14P25 #30C15 #60G15 #60G60 #Complex Variables (math.CV) #FOS: Mathematics #Geology and Paleoclimatology Research #Geometry and complex manifolds #Primary #Probability (math.PR) #math.CV #math.PR #msc:14P05 #msc:14P25 #msc:30C15 #msc:60G15 #msc:60G60 #secondary
paper · pdf · doi:10.48550/arxiv.1902.08424
30 pages
arxiv created 2019/02/22 · openalex publication_date 2019/02/22 · arxiv updated 2019/02/25 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absolute value of certain random, SO(3)-invariant rational function of degree n equals to 1. We show that the expected number of the connected components of these lemniscates, divided by n, converges to a positive constant defined in terms of the quotient of two independent plane Gaussian analytic functions. A major obstacle in applying the novel non-local techniques due to Nazarov and Sodin on this problem is the underlying non-Gaussianity, intristic to the studied model.