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

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Research Article

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


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.

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