Abstract
The convergence of the expectations of Betti numbers of Čech complexes built on binomial point processes in the thermodynamic regime is established.
Highlights
Br(x) = y ∈ Rd : y − x ≤ r denotes a ball of radius r and center x, and x is the Euclidean norm of x
The aim of this paper is to refine a limit theorem in the thermodynamic regime, a regime that n1/drn → r ∈ (0, ∞)
Note that the kth Betti number of the Cech complex built on a finite set of points in Rd is vanishing, if k ≥ d
Summary
Note that the kth Betti number of the Cech complex built on a finite set of points in Rd is vanishing, if k ≥ d. To establish a limit theorem for Betti numbers, we exploit the following two properties. For a Borel set A, let (A) denote the number of points in A.
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