Inradius of random lemniscates

TL;DR

This paper investigates the inradius of lemniscates defined by random polynomials, revealing conditions for positive lower bounds, deterministic limits, and distributional convergence, with applications to random matrix characteristic polynomials.

Contribution

It provides new probabilistic results on the inradius of random lemniscates, including bounds, limits, and distributional behavior under various zero distributions.

Findings

Inradius bounded below by a positive constant with high probability when the negative set of the logarithmic potential is non-empty.

Inradius converges to a deterministic limit if the negative set of the potential contains the support of the zeros.

Inradius of lemniscates from uniform zeros on the unit circle converges in distribution to a variable in (0,1/2).

Abstract

A classically studied geometric property associated to a complex polynomial $p$ is the inradius (the radius of the largest inscribed disk) of its (filled) lemniscate $\Lambda := \{z \in \mathbb{C}:|p(z)| < 1\}$. In this paper, we study the lemniscate inradius when the defining polynomial $p$ is random, namely, with the zeros of $p$ sampled independently from a compactly supported probability measure $\mu$. If the negative set of the logarithmic potential $U_{\mu}$ generated by $\mu$ is non-empty, then the inradius is bounded from below by a positive constant with overwhelming probability. Moreover, the inradius has a determinstic limit if the negative set of $U_{\mu}$ additionally contains the support of $\mu$. On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then the inradius converges in distribution to a random variable taking values in $(0,1/2)$. We also consider the characteristic polynomial of a Ginibre random matrix whose lemniscate we show is close to the unit disk with overwhelming probability.