Inradius of random lemniscates

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 $\Lambda ≔\{z\in ℂ:\left|p\left(z\right)\right|<1\}$.

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 (as the degree $n$ of $p$ tends to infinity). Moreover, the inradius has a deterministic limit if the negative set of $U_{\mu }$ additionally contains the support of $\mu$.

We also provide conditions on $\mu$ guaranteeing that the lemniscate is contained in a union of $n$ exponentially small disks with overwhelming probability. This leads to a partial solution to a (deterministic) problem concerning the area of lemniscates posed by Erdös, Herzog, and Piranian.

On the other hand, when the zeros are sampled independently and uniformly from the unit circle, then we show that 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.