Year: 2026
Type: preprint
Abstract: Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.
Source: arXiv (Cornell University)
Authors: Hino, Masanori, Kanazawa, Shu
Institutions: Kyoto University, Queen Mary University of London
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221098025