Limiting distribution of stabilization time

Determine the limiting behavior of the stabilization time T_n for a uniformly random permutation in S_n under an appropriate normalization.

Background

For a permutation π in S_n, the stabilization time T_n(π) is the least nonnegative integer k such that Θ^k(π) is a fixed point of the Lehmer-code operator Θ. The paper establishes a worst-case upper bound of n−1 for permutations and reports computational evidence suggesting that the mean and median stabilization times have comparable power-law growth, with an exponent near 3/4.

Beyond the expected stabilization time, the authors explicitly leave unresolved the distributional behavior of T_n. The open question is to identify an appropriate normalization and determine whether the normalized stabilization times converge in distribution, and if so, to characterize the limiting law.