Variance of the maximal degree in recursive random trees

Determine the asymptotic variance of the maximal vertex degree δ_max in the recursive random tree (RRSC(1)) model and establish whether it converges to a finite limit as N tends to infinity.

Background

The paper states that the maximal degree in the recursive random tree has a distribution sharply concentrated around a logarithmic scale, but distinguishes this distribution from the asymptotically Gaussian distribution of the primordial vertex degree.

Although the authors expect the variance of δ_max to remain finite, they explicitly identify the variance as unknown.

References

The variance $\text{Var}[\delta_\text{max}]=\langle!\langle \delta_\text{max}2\rangle!\rangle$ is unknown, but expected to remain finite in the $N\to\infty$ limit.

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Section 4, subsection “RRSC(1)”