if for allε >0, asn→ ∞,
P(|Xn−X|> ε)→0.
This convergence is also known as theweak law of large numbers for {Xn}. Definition 1.35 (Convergence in expectation). Xn converges to a ∈ R in ex-pectation, if asn→ ∞,
E[Xn]→a.
Definition 1.36 (Convergence in distribution). Xn converges toX in distribu-tion, denoted byXn−→D X, if asn→ ∞,
Fn(x)→F(x), for allx∈Rat which F(x) is continuous.
This convergence is also known as theweak convergence.
1.3 Manifold theory
In this section, we collect the basic theory of manifolds that is required in our work, from [45]. We begin with the definition of a C1 manifold and its boundary. Let Hm denote the upper half space inRm, consisting ofx∈Rm for which themth coordinate xm≥0.
Definition 1.37. A nonempty subsetMof RN, endowed with the subset topology, is called anm-dimensional C1 manifold if for eachz∈ M there exist an open setM ofMcontainingz, a setV that is open in eitherRm orHm, and aC1 bijective map ϕ:V →M such that
(i) ϕ−1:M →V is continuous;
(ii) the Jacobian ofϕ atx, denoted byJϕ(x), has rank mfor all x∈V. The pair (V, ϕ) is called a chart for z.
Definition 1.38. Let (V, ϕ) be a chart for z∈ M. We sayz is a boundary point of MifV is open inHm andz=ϕ(x) for x∈Rm−1× {0}.
We now mention what is the meaning of the integral of a measurable function κ over M. In [45], κ is assumed to be continuous, however, that can be easily generalized to measurable functions. First, consider the case where the support ofκ can be covered by a single chart. Let (V, ϕ) be that chart. Since the support ofκ is compact and ϕ−1 is continuous, without loss of generality, we can assume that V is bounded. Then the integral ofκ overMis defined as
∫
Mκ(z)dz =
∫
V◦
κ(ϕ(x))Dϕ(x)dx.
Here, V◦ = V if V is open in Rm, otherwise V◦ = V ∩Hm+, where Hm+ consists of x ∈ Rm for which xm > 0, and Dϕ(x) = det((Jtϕ(x)Jϕ(x))1/2), where Jtϕ(x) is the transpose of the Jacobian of ϕ at x. The above integral is well-defined in the sense that it is independent of the choice of chart.
Now, to define the integration in general, a concept of partition of unity is required [45, Lemma 25.2]. Since that definition (the definition following [45, Lemma 25.2]) is suitable for theoretical purposes only, instead of mentioning it, we shall use the following lemma to carry out the integration of a measurable function κoverM. Definition 1.39. A subset K of Mis said to have measure zero inM if it can be covered by countably many chartsϕi:Vi →Mi such that the set
Ki=ϕ−i 1(K∩Vi) has measure zero in Rm for each i.
Lemma 1.40([45, Theorem 25.4]). Suppose (Vi, ϕi), fori= 1,2, . . . , l, is a chart on M, such thatVi is open inRm andMis the disjoint union of open setsϕ1(V1), ϕ2(V2), . . . , ϕl(Vl) of M and a setK of measure zero in M. Then
∫
Mκ(z)dz=∑
i∈I
∫
Vi
κ(ϕi(x))Dϕi(x)dx.
The set {(Vi, ϕi), i ∈ {1,2, . . . , α}} is also called an atlas for M. Lemma 1.40 gives us a way to calculate the integral of a measurable function over M when the support of the function is covered by many charts.
Literature in brief
2.1 Introduction
In this chapter, we survey the emerging research area of random geometric complexes.
This is not the first time that one is surveying the literature on random geometric complexes. Indeed, according to our knowledge, two surveys have been written and the recent one is by Bobrowski and Kahle [8]. However, we still survey the literature for two main reasons. The first one is that many new results are appeared recently in this field, which ofcourse are not included in [8] since they appeared after the publication of [8]. So we would like to give the overall view of the results known to date for the sake of completeness. The second and the most important reason for us is that after reading this chapter, one can see very clearly where the contribution of this thesis, which is discussed in Chapters 3and 4, fit into the existing literature.
This chapter contains results related to the asymptotic behavior of (i) Betti num-bers of random geometric complexes; (ii) critical points of a distance function for point processes and Euler characteristic of random geometric complexes; (iii) persis-tent Betti numbers and persistence diagrams of the filtrations of random geometric complexes. By a random geometric complex, we mean either a random ˇCech complex or a random Vietoris-Rips complex. Random geometric complexes are built on ran-dom points with non-ranran-dom radius. The source of ranran-dom points may come from a stationary point process on RN or from a binomial point process and its Poissonized version. The binomial point process and its Poissonized version are further classi-fied into three cases depending on the support of the underlying distribution. Let Xn={X1, X2, . . . , Xn} be a binomial point process.
(i) The first case is where the common distribution ofXi has a probability density functionf(x) with respect to Lebesgue measure onRN. We shall refer to this case as theEuclidean setting.
(ii) The second case is where the common distribution is supported on a manifold M ⊂RN of dimensionm < N and has a probability density functionκ(z) with respect to the volume formdzonM. It means that ifXiis aRN-valued random
29
30 2.1. INTRODUCTION variable having density κ then for everyA⊂RN,
P(Xi ∈A) =
∫
A∩Mκ(z)dz,
where dz is a volume form on M. We shall refer to this case as the manifold setting. In the literature, M ⊂RN is assumed to be a m-dimensional smooth closed manifold, i.e., compact and without a boundary.
(iii) The final case is where the common distribution is supported on am-dimensional smooth compact Riemannian manifold (M, g), and has a probability density functionκ(z) with respect to the volume elementµg(dz), where the measureµg
is determined by g.
The main difference between the manifold setting and the case of Riemannian man-ifolds is that in the former, N-dimensional Euclidean distance is used to construct simplices of a geometric complex while in the latter, Riemannian distance is used, which is determined byg via geodesics. In the Euclidean setting, we denote the bi-nomial point process and its Poissonized version byXn and Pn respectively, while in the manifold setting as well as in the case of Riemannian manifolds, we denote them byZn and Qn respectively.
Random geometric complexes are higher-dimensional analogues of random geo-metric graphs. A random geogeo-metric graphG(Φn, r) is constructed by first choosingn random points Φnin a metric space, which becomes the vertex set of the graph. Then an edge is put between any two vertices if and only if the distance between them is less than or equals to 2r. There is a huge literature available on random geometric graphs (see for example, Penrose’s monograph [53]). While studying the asymptotic behavior of the properties (such as connectivity) of a random geometric graph, the threshold parameterr is a function of n, denoted by rn, that usually goes to zero as n→ ∞. Moreover, there exist three main limiting regimes: sparse regime, thermody-namic regime and dense regime in which the asymptotic behavior is totally different.
The same applies to random geometric complexes as well (see Figure 2.1). In the Euclidean setting, the three regimes are divided according to the limit of {n1/Nrn}:
zero, finite or infinite respectively, whereas in the manifold setting or in the case of Riemannian manifolds, they are divided according to the limit of{n1/mrn}.
Consider kth Betti number of random ˇCech complexes, which are N-valued ran-dom variables, for the illustration of basic setups in different settings. Letβk(C(Xn, rn)) be thekth Betti number of ˇCech complexC(Xn, rn) of radiusrnbuilt on the binomial point process Xn in Euclidean setting. Then in this setting, we are interested in the limiting behavior ofβk(C(Xn, rn)) as n → ∞, with the conditions that rn → 0 and n1/Nrn converges to zero, finite or infinite (depending on rn). The basic setup in the manifold setting or in the case of Riemannian manifolds is similar to this setup except now the limiting behavior of the term n1/mrn has to be taken into account instead of n1/Nrn. The basic setup in case of stationary point processes on RN is a
(a) Sparse regime (b) Thermodynamic (c) Dense regime Figure 2.1: Illustration of different limiting regimes by constructing ˇCech complexes C(X100, r100) withr100= 0.03 in (a), 0.06 in (b) and 0.1 in (c), whereX100 is a set of 100 points drawn uniformly from [0,1]×[0,1]⊂R2.
little different from that in Euclidean and manifold settings. For example, let Φ be a simple stationary point process onRN. Define
ΦL:= Φ∩(
− (L
2 )1/N
, (L
2
)1/N]N
= Φ∩WL, i.e.,
ΦLis the restriction of Φ onWL⊂RN. Then we are interested in the limiting behavior ofkth Betti number of ˇCech complexes based on ΦL, denoted by βk(C(ΦL, rL)), as L goes to infinity. Here,rL is a sequence of positive non-random radius, whose limiting behavior decides the limiting regime. In other words, the limiting regimes: sparse, thermodynamic and dense are divided according as rL (or rLN) tends to zero, finite and infinite respectively.
The above basic setups in different settings remain same when dealing with the asymptotic behavior of critical points and Euler characteristic. However, it does not make sense to have such limiting regimes for the asymptotic behavior of persistent Betti numbers (or persistence diagrams). We consider the random filtrations built on either scaled binomial point processesn1/NXn and their Poissonized versionsn1/NPn
in Euclidean setting (similarly n1/mZn and n1/mQn in manifold setting and in case of Riemannian manifolds) or on ΦLin case of stationary point processes to study the asymptotic behavior of persistent Betti numbers asn→ ∞ orL→ ∞, respectively.
The chapter is organized into three sections which contains results related to Betti numbers, critical points and Euler characteristic, persistent Betti numbers and per-sistence diagrams, respectively. We conclude this section with the following remark.
Remark 2.1. Penrose’s monograph [53] usually contains the results about the random geometric graphs in the Euclidean setting except in the last chapter, where SLLN for some quantity when the points live on torus is discussed. As we shall see in this thesis, some results for random geometric graphs (by studying them for random geometric complexes) are also extended to the case of compact manifolds with or without a boundary [58, 10] or to the case of compact Riemannian manifolds [11]. However,
32 2.2. BETTI NUMBERS