发表机构
Department of Mathematics, Graduate School of Science, Kyoto University; School of Mathematical Sciences, Queen Mary University of London（京都大学理学研究科数学系; 伦敦大学玛丽皇后学院数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对基于Costa-Farber模型的多参数复过程，建立其持续Betti数的大数定律，证明持续图依概率弱收敛到确定性有限测度，并推导了相关测度在最优传输距离下的收敛性。
AI 中文摘要
基于Costa-Farber模型的多参数复过程为增长随机单纯复形提供了统一框架，涵盖Linial-Meshulam过程和随机旗复形过程。我们建立了这些过程的持续Betti数的大数定律，并证明在确定性时间变换和归一化后，其持续图依概率弱收敛到确定性有限测度。该测度的象限质量由归一化持续Betti数的通用极限决定，该极限由显式有限维优化问题刻画。我们将图收敛扩展到死亡坐标具有多项式增长且对角处具有受控奇异性的连续函数的积分。这些结果得出加性和乘性寿命测度及总持续性的收敛，进一步推导出最优传输距离下的收敛。证明结合了单纯复形对的局部弱收敛、考虑共享面依赖的压缩拉普拉斯算子的谱分析，以及大死亡坐标和短寿命的定量估计。
英文摘要
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.
Comments78 pages, 4 figures
