Inradius of random lemniscates
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 nonempty, 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2023
 DOI:
 10.48550/arXiv.2301.13424
 arXiv:
 arXiv:2301.13424
 Bibcode:
 2023arXiv230113424K
 Keywords:

 Mathematics  Probability;
 Mathematics  Complex Variables;
 30C10;
 60G60;
 31A15
 EPrint:
 21 pages, 4 figures