On the topology of random complexes built over stationary point processes

Yogeshwaran, D.; Adler, Robert J.

doi:10.1214/14-aap1075

Cited by 51 publications

(76 citation statements)

References 41 publications

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“…Theorem 1.12 shows these limits hold for general stationary input. The paper [74] gives a weaker version of Theorem 1.12 for specific ξ and for f = 1[x ∈ W 1 ]. In full generality, the convergence rate (1.23) is new.…”

Section: Remarksmentioning

confidence: 99%

Limit theory for geometric statistics of point processes having fast decay of correlations

Błaszczyszyn

Yogeshwaran²,

Yukich³

2019

Ann. Probab.

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Let P be a simple, stationary point process on R d having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let Pn := P ∩ Wn be its restriction to windows Wn := [− 1 2 n 1/d , 1 2 n 1/d ] d ⊂ R d . We consider the statistic H ξ n := x∈Pn ξ(x, Pn) where ξ(x, Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for H ξ n and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures µ ξ n := x∈Pn ξ(x, Pn)δ n −1/d x , as Wn ↑ R d . This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ N) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [72] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [53] and Shirai and Takahashi [71], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [12,13] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the asymptotic normality of µ ξ n via an extension of the cumulant method.

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Section: Remarksmentioning

confidence: 99%

Limit theory for geometric statistics of point processes having fast decay of correlations

Błaszczyszyn

Yogeshwaran²,

Yukich³

2019

Ann. Probab.

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“…We will refer to these generalizations as random combinatorial complexes (see [26] for a survey). It turns out that the Erdős -Rényi threshold for connectivity can be generalized to that of "homological connectivity," where the higher homology groups H k become trivial.In parallel to the study of combinatorial complexes, a line of research was established for random geometric complexes [3,5,24,42,43]. This type of complexes generalizes the model of the random geometric graph G(n, r) (introduced by Gilbert [18]), where vertices are placed at random in a metric-measure space, and edges are included based on proximity [38].…”

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confidence: 99%

Random Čech complexes on Riemannian manifolds

Bobrowski

Oliveira

2018

Random Struct Algorithms

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In this paper we study the homology of a random Čech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.

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“…Further justification for negative association as a measure of sparsity was seen in [37] where it was shown that the Palm measure of a negatively associated PP is 'stochastically weaker' than that of the original PP. In particular, the void probability increases for the Palm measure.…”

Section: Comparison Of Clustering To Poisson Point Processesmentioning

confidence: 99%

“…The motivation lies in the recent subject of topological data analysis. In an upcoming work [37], we study these models on more general stationary PPs using tools of stochastic ordering as well as asymptotic analysis of joint intensities and void probabilities. In particular, if we denote r con n ( ) as the critical contractibility radius for theCech complex on ∩ [−n 1/d /2, n 1/d /2] (i.e.…”

Section: First Passage Percolationmentioning

confidence: 99%

On Comparison of Clustering Properties of Point Processes

Błaszczyszyn

Yogeshwaran

2014

Adv. Appl. Probab.

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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 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. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

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On the topology of random complexes built over stationary point processes

Cited by 51 publications

References 41 publications

Limit theory for geometric statistics of point processes having fast decay of correlations

Limit theory for geometric statistics of point processes having fast decay of correlations

Random Čech complexes on Riemannian manifolds

On Comparison of Clustering Properties of Point Processes

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