Central limit theorem for the giant k-deeply connected component

Determine whether a central limit theorem holds for the size of the giant k-deeply connected component in the uniform and/or binomial random graph models.

Background

The paper proves a law-of-large-numbers-type result for the size of the giant k-deeply connected component: above the threshold, its size is asymptotically given by a function determined by the largest solution of a Poisson fixed-point equation.

The authors do not derive fluctuation limits for this size. They explicitly ask whether a central limit theorem holds in either of the two standard sparse random graph models considered in the paper, namely the uniform model and the binomial model.