2023/01/31 by Manjunath Krishnapur, Erik Lundberg, Krishnapur, Manjunath +3
Mathematics · #Point processes and geometric inequalities #Random Matrices and Applications #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2301.13424
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 ∈ 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 Uμ 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 Uμ 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.