  We determine the true asymptotic behaviour for the expected number of
connected components for a model of random lemniscates proposed recently by
Lerario and Lundberg. These are defined as the subsets of the Riemann sphere,
where the absolute value of certain random, $\text{SO}(3)$-invariant rational
function of degree $n$ equals to $1$. We show that the expected number of the
connected components of these lemniscates, divided by $n$, converges to a
positive constant defined in terms of the quotient of two independent plane
Gaussian analytic functions. A major obstacle in applying the novel non-local
techniques due to Nazarov and Sodin on this problem is the underlying
non-Gaussianity, intristic to the studied model.
