Law of large numbers for Betti numbers of homogeneous and spatially independent random simplicial complexes

Enumerating the class of minimally path connected simplicial complexes

In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of being homologically $\Bbb Q$-acyclic. In this text we consider a different generalisation of trees to the class of pure dimensional simplicial complexes that \textit{minimally connect a vertex set} (in the sense that the removal of any top dimensional face disconnects the complex). Our main result provides an upper and lower bound for the number of these mi…

Topology of random simplicial complexes: a survey

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.

Gigantic random simplicial complexes

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.

Large deviations for subcomplex counts and Betti numbers in multi-parameter simplicial complexes

We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erdös-Rényi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large deviations. We first study upper tail large deviation probabilities for subcomplex counts, deriving the order of magnitude of such probabilities at the logarithmic scale precision. The obtained results are then applied to analyze large deviations for the number of simplices at the critical dimension and below. Finally, these results are also used to deduce larg…

多参数随机单纯复形中高维 Betti 数的极限定理

Limit theorems for high-dimensional Betti numbers in the multiparameter random simplicial complexes

本文研究了多参数随机单纯复形在临界维度以上的拓扑性质，证明了高维 Betti 数满足强大数定律与中心极限定理，并探讨了下尾大偏差及其相变现象。

软随机单纯复形的中心极限定理

Central limit theorems for Soft random simplicial complexes

本文提出了建立在软随机图上的软随机单纯复形模型，并利用正态逼近定理证明了其 $k$-面数量与欧拉示性数的中心极限定理。

Random Complexes and l^2-Betti Numbers

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin.

高维单纯复形与布尔模型的随机连接模型中的Betti数

Betti numbers in the Random Connection Model for higher-dimensional simplicial complexes and the Boolean model

本文研究arXiv:2506.11918提出的推广随机连接模型的新型随机单纯复形模型，推导了包含Betti数在内的泛函的中心极限定理，给出布尔模型中Betti数的中心极限定理。

Random Walks on Simplicial Complexes and Harmonics

In this paper, we introduce random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension $d$, a random walk with an absorbing state is defined which relates to the spectrum of the $k$-dimensional Laplacian for $1 \leq k \leq d$ and which relates to the local random walk on a graph defined by Fan Chung. We also examine an application of random walks on simplicial complexes to a semi-supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classificati…

Local-to-Global Contraction in Simplicial Complexes

We give a local-to-global principle for relative entropy contraction in simplicial complexes. This is similar to the local-to-global principle for variances obtained by Alev and Lau (2020).