Survival probability of the leading degree in recursive random trees

Determine the asymptotic behavior of the probability Φ(N) that the primordial vertex in the recursive random tree (RRSC(1)) model has degree at least as large as every other vertex at every stage of the evolution up to N vertices.

Background

The quantity Φ(N) records whether the primordial vertex remains a maximum-degree vertex throughout the entire growth history, rather than only at the final time. The paper contrasts this with the probability F(N) of being maximal only at the end, whose asymptotic behavior is known for the recursive random tree.

Even in the RRSC(1)=RRT case, the authors report that the survival probability has not been determined.

References

Even for the RRSC(1), i.e., the RRT, we only know the probability distribution $P(\delta_1,N)$ and the asymptotic behavior of $F(N)$; the `survival' probability $\Phi(N)$ is still unknown.

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