Homological connectivity in random Čech complexes

Poisson process approximation under stabilization and Palm coupling

“…Grouping together all critical points of the same index, we have a point process for which we wish to prove Poisson convergence under suitable scaling. This convergence statement has a significant contribution to the analysis of the homological connectivity phenomenon studied in [Bob22]. In particular, it yields the asymptotic behaviour of the persistent homology in the critical window for homological connectivity (see Remark 6.3).…”

“…Critical points of index k can then affect the homology of B r (η) either in dimension k (creating new k-dimensional cycles) or in dimension k − 1 (terminating a (k − 1)-dimensional cycle). Thus, the work in [BA14,Bob22] focused on the critical points and their indexes, as a proxy to the homology. We shall define critical points more formally below.…”

“…The work in [Bob22] focused on a homogeneous Poisson process defined on a flat torus X = T d = R d /Z d (which can be thought of as the unit box [0, 1] d with a periodic boundary). Specifically, it considered a Poisson process η = η n on T d with a fixed n. The main objective there was to characterize the phase transition for homological connectivity, i.e., where the k th homology of the random union B r (η n ) converges to the k th homology of the underlying torus T d .…”

“…Let R n be any value satisfying (6.2) lim n → ∞ R n = 0, and lim n → ∞ r n /R n = 0. The analysis in [Bob22] focused on the number of critical points c ∈ T d with Morse index k, such that ρ(c; η n ) ∈ (r n , R n ]. Defining by C k,n the number of such critical points, in [Bob22, Proposition 4.1] the following was proved.…”

“…Note that the limiting results are independent of the choice of R n (provided that (6.2) is satisfied). To prove Proposition 6.1, [Bob22] used the fact that critical points of index k are generated by subsets x ∈ (η n ) k+1 ̸ = . Defining…”