On the topology of random complexes built over stationary point processes.

@article{Yogeshwaran2015OnTT,
  title={On the topology of random complexes built over stationary point processes.},
  author={D. Yogeshwaran and Robert J. Adler},
  journal={Annals of Applied Probability},
  year={2015},
  volume={25},
  pages={3338-3380}
}
Cech and Vietoris-Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either iid observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction, or repulsion, we find phenomena quantitatively different from those… 
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References

SHOWING 1-10 OF 45 REFERENCES
Random geometric complexes in the thermodynamic regime
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the…
Random Geometric Complexes
  • Matthew Kahle
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 2011
TLDR
The expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd are studied and asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes are given.
On Comparison of Clustering Properties of Point Processes
In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that…
Topology of Random 2-Complexes
TLDR
This paper studies the Linial–Meshulam model of random two-dimensional simplicial complexes and proves that for p≪n−1 a random 2-complex Y collapses simplicially to a graph and, in particular, the fundamental group π1(Y) is free and H2(Y)=0, asymptotically almost surely.
The fundamental group of random 2-complexes
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in…
Limit theorems for Betti numbers of random simplicial complexes
There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases…
Clustering and percolation of point processes
We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller…
Topology of random clique complexes
TLDR
Estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.
Statistical topology via Morse theory, persistence and nonparametric estimation
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators…
Distance Functions, Critical Points, and Topology for Some Random Complexes
For a finite set of points $P$ in $R^d$, the function $d_P:R^d \to R_+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is random. In particular,…
...
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