A remark on the convergence of Betti numbers in the thermodynamic regime
 Khanh Duy Trinh^{1}Email author
https://doi.org/10.1186/s4073601700290
© The Author(s) 2017
Received: 14 November 2016
Accepted: 27 February 2017
Published: 6 March 2017
Abstract
The convergence of the expectations of Betti numbers of Čech complexes built on binomial point processes in the thermodynamic regime is established.
Keywords
AMS Subject Classification
Terminologies and main results
Definition 1.1

The 0simplices (vertices) are the points in \(\mathfrak {X}\).

A ksimplex \(\left [x_{i_{0}}, \dots, x_{i_{k}}\right ]\) is in \(\mathcal {C}(\mathfrak {X}, r)\) if \(\bigcap _{j = 0}^{k} B_{r} (x_{i_{j}}) \ne \emptyset \).
Here \(B_{r}(x) = \left \{y \in \mathbb {R}^{d} : \y  x\ \le r\right \}\) denotes a ball of radius r and center x, and ∥x∥ is the Euclidean norm of x. The Čech complex can be also constructed from an infinite collection of points.
Let X _{1},X _{2},…, be a sequence of i.i.d. (independent identically distributed) \(\mathbb {R}^{d}\)valued random variables with common probability density function f(x). Define the induced binomial point processes as \(\mathfrak {X}_{n} = \left \{X_{1}, \dots, X_{n}\right \}\). The object here is the Čech complex \(\mathcal {C}\left (\mathfrak {X}_{n}, r_{n}\right)\) built on \(\mathfrak {X}_{n}\), where the radius r _{ n } also varies with n. Denote by \(\beta _{k}(\mathcal {K})\) the kth Betti number, or the rank of the kth homology group, of a simplicial complex \(\mathcal {K}\). The limiting behaviour of Betti numbers \(\beta _{k}\left (\mathcal {C}\left (\mathfrak {X}_{n}, r_{n}\right)\right)\) in various regimes has been studied recently by many authors. See [1] for a brief survey. The aim of this paper is to refine a limit theorem in the thermodynamic regime, a regime that n ^{1/d } r _{ n }→r∈(0,∞).
provided that the density function f has compact, convex support and that on the support of f, it is bounded both below and above [9, Theorem 4.6]. A remaining problem is to describe the exact limiting behaviour of the expected values of the Betti numbers. This paper gives a solution to that problem. Note that the 0th Betti number which counts connected components in a random geometric graph was completely described [5, Chapter 13]. Note also that the kth Betti number of the Čech complex built on a finite set of points in \(\mathbb {R}^{d}\) is vanishing, if k≥d. These facts explain why we only need to consider the case 1≤k≤d−1.
To establish a limit theorem for Betti numbers, we exploit the following two properties. The first one is the nearly additive property of Betti numbers that was used in [9] to study Betti numbers of Čech complexes built on stationary point processes. The second one is the property that binomial point processes behave locally like a homogeneous Poisson point process. The latter property is also a key tool to establish the law of large numbers for local geometric functionals [6, 7].
Definition 1.2

for disjoint Borel sets A _{1},…,A _{ k }, the random variables \(\mathcal {P}(A_{1}), \dots, \mathcal {P}(A_{k})\) are independent;

for any bounded Borel set A, the number of points in A has Poisson distribution with parameter A, \(\mathcal {P}(A) \sim \text {Pois}~(\lambda A)\), that is,where A denotes the Lebesgue measure of A.$$ \mathbb{P}(\mathcal{P}(A) = k) = e^{ \lambda A} \frac{\lambda^{k} A^{k}}{k!}, \quad k = 0,1, \dots, $$
Theorem 1.3
Consequently, together with (1), we have the following law of large numbers.
Corollary 1.4
It is noted that the method here can be applied to show the convergence of persistence diagrams of Čech complexes built on binomial point processes. The convergence of Betti numbers and persistence diagrams related to i.i.d. sampling were observed in [4] by numerical simulation. Here we give a rigorous mathematical proof of the convergences.
Definition 1.5

for disjoint Borel sets A _{1},…,A _{ k }, the random variables \(\mathcal {P}\left (A_{1}\right), \dots, \mathcal {P}\left (A_{k}\right)\) are independent;

for any bounded Borel set A, \(\mathcal {P}(A)\sim \text {Pois}\left (\int _{A} f(x) dx\right)\).
As proved later, Theorem 1.3 is equivalent to the following result.
Theorem 1.6
All the proofs will be given in Section 3 after discussing some basic properties of Betti numbers in the next section.
Simplicial complexes and Betti numbers
This section introduces some basic concepts in algebraic topology such as simplicial complexes and Betti numbers. It is mainly taken from the book [3].
An abstract simplicial complex \(\mathcal {K}\) on a finite set V is a collection of nonempty subsets of V which is closed under inclusion relation, that is, if \(\sigma \in \mathcal {K}\), then \(\tau \in \mathcal {K}\) for any nonempty subset τ⊂σ. An element \(\sigma \in \mathcal {K}\) with σ=k+1 is called a ksimplex or a simplex of dimension k. A 0simplex (resp. 1simplex) is usually called a vertex (resp. edge). Čech complexes are examples of geometric complexes which are constructed over points in some metric space with respect to certain conditions.
We assign orientations on simplices in the following way. For a ksimplex σ={v _{0},…,v _{ k }} with k>0, define two orderings of its vertex set to be equivalent if they differ from one other by an even permutation. The orderings of the vertices of σ then fall into two equivalent classes. Each of these classes is called an orientation of σ. We write 〈v _{0},…,v _{ k }〉 for an oriented simplex. Let us fix an ordering of the vertex set V. Then the notation 〈σ〉 means the oriented simplex which belongs to the equivalent class of a natural ordering. A 0simplex has only one orientation.
Lemma 2.1
We have mentioned that Betti numbers are nearly additive because of the two properties (4) and (5). Note that β _{0} counts the number of connected components in the undirected graph G=(V,E), where \(E =\mathcal {K}_{1}\), which is independent of the underlying field F.
Proofs of main theorems

Scaling property. For any θ>0 and \(t \in \mathbb {R}^{d}\),where ‘\(\overset {d}{=}\)’ denotes the equality in distribution. In particular, \(\theta (\mathcal {P}(\lambda)  t) \overset {d}{=} \mathcal {P}(\theta ^{d} \lambda)\).$$ \theta (\mathcal{P}(f(x))  t) \overset{d}{=} \mathcal{P}(\theta^{d} f(t + \theta^{1}x)), $$

Coupling property. Let \(\mathcal {P}(g(x))\) be a Poisson point process with intensity function g(x) which is independent of \(\mathcal {P}(f(x))\). ThenHere ‘ + ’ means the superposition of two point processes.$$ \mathcal{P}(f(x)) + \mathcal{P}(g(x)) \overset{d}{=} \mathcal{P}(f(x) + g(x)). $$
We begin with a result for the simplices counting function.
Lemma 3.1
Proof
For the sake of simplicity, we denote by β _{ k }(λ,r;L) the kth Betti number of the Čech complex \(\mathcal {C}(\mathcal {P}_{W_{L}} (\lambda), r)\), where W _{ L } is any rectangle of the form \(x + [\frac {L^{1/d}}{2}, \frac {L^{1/d}}{2})^{d}\).
Lemma 3.2
In particular, \(\hat \beta _{k}(\lambda, r) = \lambda \hat \beta _{k} \left (1, \lambda ^{1/d} r\right)\) is a continuous function in both λ and r, and \(\hat \beta (\lambda, r) > 0\), if λ>0 and r>0.
Proof
The sequence \(\left \{L^{1}\mathbb {E}\left [S_{j}(\lambda, r; L)\right ]\right \}\) converges uniformly on [ 0,Λ] by Lemma 3.1, and hence, is equicontinuous, which then implies the equicontinuity of the sequence \(\left \{L^{1} \mathbb {E}\left [ \beta _{k} (\lambda, r; L)\right ]\right \}\).
By observing that \(\theta ^{1/d} \mathcal {P}(\lambda)\) has the same distribution with \(\mathcal {P}(\lambda \theta)\), we obtain the scaling property of \(\hat \beta _{k}(\lambda, r)\). It then follows from the scaling property that \(\hat \beta _{k}(\lambda, r)\) is continuous in both λ and r. The lemma is proved. □
Theorem 3.3
Remark 3.4.
Note that \(\mathcal {P}'_{n} = \left (r/\tilde {r}_{n}\right) \mathcal {P}_{n}\) is also a nonhomogeneous Poisson point process. Moreover, as a result of scaling, \(\mathcal {C}\left (\mathcal {P}_{n}, \tilde {r}_{n}\right) \cong \mathcal {C}\left (\mathcal {P}_{n}', r\right)\). Thus it is enough to prove Theorem 3.3 with \(\tilde {r}_{n} = r\).
Lemma 3.5
Proof
An analogous estimate holds when we compare the kth Betti number of \(\mathcal {C}(\mathcal {P}(g(x)), r)\) and that of \(\mathcal {C}(\mathcal {P}(h(x)), r)\). The proof is complete. □
Proof of Theorem 3.3
Let S be the support of f and Λ:= supf(x). Divide \(\mathbb {R}^{d}\) according to the lattice \((L/n)^{1/d}\mathbb {Z}^{d}\) and let {C _{ i }} be the cubes which intersect with S. Since we also consider the Poisson point process with density 0, we may assume that S=∪_{ i } C _{ i }.
Here \(S_{j}\left (\mathcal {P}_{n}, r; A\right)\) is the number of jsimplices in \(\mathcal {C}\left (\mathcal {P}_{n}, r\right)\) which has a vertex in A, ∂ A denotes the boundary of the set A and A ^{(2r)} is the set of points with distance at most 2r from A. The second inequality holds because any simplex in \(\mathcal {C}\left (\mathcal {P}_{n}, r\right) \setminus \cup _{i}\mathcal {C}(\mathcal {P}_{n}_{W_{i}}, r)\) must have a vertex in ∪_{ i }(∂ W _{ i })^{(2r)}.
because the function f(x) is assumed to be Riemann integrable.
Combining the two estimates (14) and (15) and then let L→∞, we get the desired result. The proof is complete. □
The result for binomial point processes will follow from Theorem 1.6 and the following result.
Lemma 3.6
Proof
Declarations
Acknowledgements
The author would like to thank Professor Tomoyuki Shirai and Dr. Kenkichi Tsunoda for useful discussions. The author would also like to thank the referees for their valuable comments. This work is partially supported by JST CREST Mathematics (15656429). The author is partially supported by JSPS KAKENHI Grant No. 16K17616.
Competing interests
The author declares that he has no competing interests.
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