General asymptotic self-averaging and Gaussianity of simplex counts

Prove, for every RRSC(m) model with m≥2 and every admissible simplex dimension d≤m, the asserted asymptotic self-averaging, linear mean and variance growth, and asymptotic Gaussianity of the random simplex counts S_d as the number of vertices N tends to infinity.

Background

The paper asserts that the numbers S_d of d-dimensional simplices have extensive means and variances and become asymptotically Gaussian, but develops the analysis in detail only for RRSC(2) and partially for RRSC(3). The authors state that these claims have not been proved in full generality for arbitrary m and d.

The missing proof is important because asymptotic self-averaging is used throughout the derivation of the amplitudes governing the linear growth of simplex counts in the RRSC(m) models. The discussion section reiterates that this foundational property remains unproved beyond the cases treated analytically.

References

We do not prove the above assertions in full generality, i.e., for all $m\geq 2$ and all $d\leq m$.

— Random Recursive Simplicial Complexes  (2608.26547 - Krapivsky et al., 27 Aug 2026) in Section 1, discussion preceding Section 2; Section 6, Discussion