Proof of non-self-averaging of extreme degrees

Prove that the primordial degree δ_1 and the maximal degree δ_max are non-self-averaging random variables for the RRSC(m) models with m≥2.

Background

The paper derives scaling estimates and numerical evidence indicating that the primordial and maximal degrees retain fluctuations comparable to their means in RRSC(2) and more generally in RRSC(m) for m≥2.

However, the claimed non-self-averaging property is not established analytically, leaving open the task of proving it for the full class of models.

References

The proof of this assertion is lacking, although the numerical evidence is quite compelling.

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Section 4, subsection “RRSC(2)”