On the geometry of random lemniscates

We investigate the geometry of a random rational lemniscate G, the level set {|r(z)| = 1} on the Riemann sphere C = C boolean OR {infinity} of the modulus of a random rational function r. We assign a probability distribution to the space of rational functions r = p/q of degree n by sampling p and q independently from the complex Kostlan ensemble of random polynomials of degree n. We prove that the average spherical length of G is (pi(2)/2)root n, which is proportional to the square root of the maximal spherical length. We also provide an asymptotic for the average number of points on the curve Gamma that are tangent to one of the meridians on the Riemann sphere (that is, tangent to one of the radial directions in the plane). Concerning the topology of G on a local scale, we prove that, for any disk D of radius n- 1/2 on the Riemann sphere and any fixed arrangement (isotopy type of finitely many embedded circles), there is a positive probability (independent of n) that G n D realizes the prescribed arrangement (a local random version of Hilbert's Sixteenth Problem restricted to lemniscates). Corollary: the average number of connected components of G increases linearly (the maximum rate possible according to a deterministic upper bound).