Asymptotics for the expected number of nodal components for random lemniscates

Zakhar Kabluchko; Igor Wigman

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arXiv · 1902.08424

Asymptotics for the expected number of nodal components for random lemniscates

Zakhar Kabluchko·Igor Wigman

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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, $\text{SO}(3)$-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.

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Zakhar Kabluchko, Igor Wigman. 2019-02-22. Asymptotics for the expected number of nodal components for random lemniscates. https://arxiv.org/abs/1902.08424

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