  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.
