Thresholds for vanishing of 'isolated' faces in random čech and vietoris-rips complexes

Article Type

Research Article

Publication Title

Annales de l'institut Henri Poincare (B) Probability and Statistics

Abstract

We study combinatorial connectivity for two models of random geometric complexes. These two models - Čech and Vietoris-Rips complexes - are built on a homogeneous Poisson point process of intensity n on a d-dimensional torus, d > 1, using balls of radius rn. In the former, the k-simplices/faces are formed by subsets of (k + 1) Poisson points such that the balls of radius rn centred at these points have a mutual interesection and in the latter, we require only a pairwise intersection of the balls. Given a (simplicial) complex (i.e., a collection of k-simplices for all k ≥ 1), we can connect k-simplices via (k + 1)-simplices ('up-connectivity') or via (k - 1)-simplices ('down-connectivity). Our interest is to understand these two combinatorial notions of connectivity for the random Čech and Vietoris-Rips complexes asymptotically as n→∞. In particular, we analyse in detail the threshold radius for vanishing of isolated k-faces for up and down connectivity of both types of random geometric complexes. Though it is expected that the threshold radius rn = Θ(( log n/n )1/d ) in coarse scale, our results give tighter bounds on the constants in the logarithmic scale as well as shed light on the possible second-order correction factors. Further, they also reveal interesting differences between the phase transition in the Čech and Vietoris-Rips cases. The analysis is interesting due to non-monotonicity of the number of isolated k-faces (as a function of the radius) and leads one to consider 'monotonic' vanishing of isolated k-faces. The latter coincides with the vanishing threshold mentioned above at a coarse scale (i.e., log n scale) but differs in the log log n scale for the Čech complex with k = 1 in the up-connected case. For the case of up-connectivity in the Vietoris-Rips complex and for rn in the critical window, we also show a Poisson convergence for the number of isolated k-faces when k ≤ d.

First Page

1869

Last Page

1897

DOI

10.1214/19-AIHP1020

Publication Date

8-1-2020

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