LX growth asymptotics

Characterize the asymptotic LX growth distribution by proving the stated bounds on asymmetry and kurtosis and determining whether the distribution converges to the Gumbel distribution.

Background

The authors report that LX growth appears asymmetric and is better fitted by a Gumbel distribution than by a Gaussian. They conjecture numerical asymptotics for skewness, kurtosis, mean, and mode, while explicitly noting uncertainty about convergence to the Gumbel law.

References

It might be also that the distribution tends to the Gumbel one when $n\to \infty$, however that is not completely clear since empiric skew and kurt are not well fitted (however it might be an effect of small $n$).

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 12, subsection “Further results and conjectures on growth”