Gumbel law for LRX growth

Prove that the growth distribution of the LRX Cayley graph of S_n converges, for large n, to a Gumbel distribution.

Background

Computations of LRX graph growth for n≤14 suggest an asymmetric distribution distinct from the Gaussian behavior associated with Coxeter-type generators. The authors formulate a central-limit-type conjecture that the growth distribution approaches a Gumbel law as n becomes large.

References

Based on explicit computations for the growth for $n \le 14$ and its analysis (notebooks \href{https://www.kaggle.com/code/ogurtsov/gumbel}{1} \href{https://www.kaggle.com/code/ogurtsov/gumbel-for-binary-puzzle}{2} ) we come to the following conjecture, which can be thought as an analogue of the central limit theorems.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 5.5, “Conjectures: Gumbel for growth, spectrum uniformity, random walks mixing, etc.”