Asymptotics for the Expected Number of Nodal Components for Random Lemniscates

DOI10.1093/IMRN/RNAA146zbMATH Open1486.30017arXiv1902.08424OpenAlexW2929110139MaRDI QIDQ5028234

Zakhar Kabluchko, Igor Wigman

Publication date: 8 February 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: 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, $extSO(3)$-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.

Full work available at URL: https://arxiv.org/abs/1902.08424

zbMATH Keywords

rational functions

Mathematics Subject Classification ID

Polynomials and rational functions of one complex variable (30C10)

Related Items (1)

On the number of components of random polynomial lemniscates

This page was built for publication: Asymptotics for the Expected Number of Nodal Components for Random Lemniscates

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q5028234)