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arXiv · 2301.13424

Inradius of random lemniscates

Manjunath Krishnapur·Erik Lundberg·Koushik Ramachandran
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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 $Λ:= \{z \in \mathbb{C}:|p(z)| < 1\}$. 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 $μ$. If the negative set of the logarithmic potential $U_μ$ generated by $μ$ 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 $U_μ$ additionally contains the support of $μ$. 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.

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Manjunath Krishnapur, Erik Lundberg, Koushik Ramachandran. 2023-01-31. Inradius of random lemniscates. https://arxiv.org/abs/2301.13424

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