# Asymptotics for the expected number of nodal components for random lemniscates

## TL;DR

This paper establishes the asymptotic behavior of the expected number of connected components in random lemniscates, showing it grows linearly with degree and converges to a positive constant, despite non-Gaussian complexities.

## Contribution

It provides the first true asymptotic analysis of the expected number of nodal components in a non-Gaussian random lemniscate model.

## Key findings

- Expected number of components divided by degree converges to a positive constant.
- The constant is expressed via Gaussian analytic functions.
- Overcoming non-Gaussian challenges with novel techniques.

## 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.