On the geometry of random lemniscates

TL;DR

This paper studies the geometric properties of random rational lemniscates on the Riemann sphere, revealing their average length, tangent point statistics, and local topological complexity, with implications for understanding their global structure.

Contribution

It provides the first detailed probabilistic analysis of the geometry and topology of random rational lemniscates, including average length, tangent points, and local isotopy behavior.

Findings

Average spherical length scales as √n

Positive probability of local topological configurations

Average number of connected components grows linearly

Abstract

We investigate the geometry of a random rational lemniscate $\Gamma$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational functions $r=p/q$ of degree $n$ by sampling $p$ and $q$ independently from the complex Kostlan ensemble of random polynomials of degree $n$. We prove that the average \emph{spherical length} of $\Gamma$ is $\frac{\pi^2}{2} \sqrt{n},$ which is proportional to the square root of the maximal spherical length. We also provide an asymptotic for the average number of points on the curve that are tangent to one of the meridians on the Riemann sphere (i.e. tangent to one of the radial directions in the plane). Concerning the topology of $\Gamma$, on a local scale, we prove that for every disk $D$ of radius $O(n^{-1/2})$ in the Riemann sphere and any \emph{arrangement} (i.e. embedding) of finitely many circles $A\subset D$ there is a positive probability (independent of $n$) that $(D,\Gamma\cap D)$ is isotopic to $( D,A)$. (A local random version of Hilbert's Sixteenth Problem restricted to lemniscates.) Corollary: the average number of connected components of $\Gamma$ increases linearly (the maximum rate possible according to a deterministic upper bound).