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arXiv · 2608.26547

Random Recursive Simplicial Complexes

P. L. Krapivsky·M. Lucas
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Abstract

We investigate random recursive simplicial complexes growing by adding, at each step, a vertex together with a simplex formed by joining the new vertex with a randomly chosen existing simplex. We also add all faces of the new simplex to ensure that the resulting object remains a simplicial complex. If the choice of an existing simplex is uniform among simplices of dimension $<m$, the number $S_d$ of simplices of any admissible dimension $d\leq m$ is an asymptotically self-averaging random variable. This feature allows us to determine the asymptotic growth law of the average of $S_d$ when the number of vertices diverges. We also probe the degree distribution, examine the probabilities of various extreme outcomes, and analyze the characteristics of the first vertex.

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P. L. Krapivsky, M. Lucas. 2026-08-27. Random Recursive Simplicial Complexes. https://arxiv.org/abs/2608.26547

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