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

@article{Yogeshwaran2012OnTT,
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={2012},
volume={25},
pages={3338-3380}
}
• Published 1 November 2012
• Mathematics
• Annals of Applied Probability
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…

## Figures from this paper

• Mathematics
Random Struct. Algorithms
• 2017
The homology of random \v{C}ech complexes over a homogeneous Poisson process on the d-dimensional torus is computed, and it is shown that there are, coarsely, two phase transitions.
• Mathematics
Probability Theory and Related Fields
• 2015
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
• Mathematics
• 2018
We establish the strong law of large numbers for Betti numbers of random \v{C}ech complexes built on $\mathbb R^N$-valued binomial point processes and related Poisson point processes in the
• Mathematics
• 2020
Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting,
• Mathematics, Computer Science
Advances in Applied Probability
• 2014
It is proved that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure.
• Mathematics
• 2022
We prove a large deviation principle for the point process associated to k -element connected components in R d with respect to the connectivity radii r n → ∞ . The random points are generated from a
• Mathematics
• 2022
We prove a large deviation principle for the point process associated to k -element connected components in R d with respect to the connectivity radii r n → ∞ . The random points are generated from a
We investigate the topological dynamics of extreme sample clouds generated by a heavy tail distribution on R by establishing various limit theorems for Betti numbers, a basic quantifier of algebraic
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such

## References

SHOWING 1-10 OF 56 REFERENCES

• Mathematics, Computer Science
Advances in Applied Probability
• 2014
It is proved that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure.
• Mathematics
Discret. Comput. Geom.
• 2012
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.
• Mathematics
• 2010
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
• Mathematics, Computer Science
• 2013
It is shown that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process.
• Mathematics
• 2009
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
• Mathematics
• 2011
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,
• Mathematics
University Lecture Series
• 2009
The book examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients, which share a property of 'point-repulsion', and presents a primer on modern techniques on the interface of probability and analysis.
Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our
A number of questions from a variety of areas of mathematics lead one to the problem of analyzing the topology of a simplicial complex. However, there are few general techniques available to aid us