Random geometric complexes in the thermodynamic regime

Abstract

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.

Appendices

Appendices

1.1 Appendix 1

The following useful lemma needed for variance lower bounds is essentially a simplification of [23, Theorem 5.2] to our situation.

Lemma 5.1

Let $n \ge 1$ and ${\mathcal {P}}_n$ be the Poisson point process with density nf,  where f satisfies (4.1). Let F be a translation invariant functional on locally finite point sets of ${\mathbb {R}}^d$ such that ${\mathbb {E}}\left\{ F({\mathcal {P}}_n)^2\right\} < \infty$. Assume that there exist $m \in {\mathbb {N}},$ a set $A \subset \mathrm{{supp}}(f),$ a finite set of points $z_1,\ldots , z_{m}$ and $r > 0$ with $A \oplus B_{z_i}(r) \subset \mathrm{{supp}}(f)$ for all $i \in \{1,\ldots ,m\}$ such that for all $x \in A$ and $(x_1,\ldots ,x_m) \in \prod _{i=1}^mB_{x+z_i}(r),$ we have that

$$\begin{aligned} \big |{\mathbb {E}}\left\{ D_x \big ( F({\mathcal {P}}_n \cup \{x_1\ldots ,x_m\}) \big )\right\} \big | \ge c, \end{aligned}$$

(5.1)

for some positive constant c. Then

$$\begin{aligned} \mathsf {VAR}\left( F({\mathcal {P}}_n)\right) \ge f^*n \frac{c^2(f_*(f^*)^{-1})^{m+1}}{8^{m+2}\cdot 4\cdot (m+1)!} \min _{j=1,\ldots ,m+1} 2^{-d(m+1-j)}(w_df^*nr^d)^{j-1}|A|.\nonumber \\ \end{aligned}$$

(5.2)

Proof

[23, Theorem 5.3] simplifies [23, Theorem 5.2] to the case of stationary Poisson point processes in Euclidean space. However, since we are dealing with Poisson point processes with non-uniform densities, we require a small change in the arguments there and shall describe this in a moment. In any case, the similarity of our lower bound (5.2) to that of [23, Theorem 5.3] is to be expected. In particular, if we set $f_* = f^* = \lambda$, then (5.2) is exactly the variance lower bound in [23, Theorem 5.3] with t there replaced by $\lambda n$ here (although the set A and r are different).

Now, more specifically, set

$$\begin{aligned} U = \left\{ (x,x+z_1+x_1,\ldots ,x+z_m+x_m) :\, x \in A, \ (x_1,\ldots ,x_m) \in \prod _{i=1}^mB_O(r) \right\} . \end{aligned}$$

Note that this plays the role of U as defined in [23, Theorem 5.2] and defining g on U as

$$\begin{aligned} g(y_1,\ldots ,y_{m+1}) := \big |{\mathbb {E}}\left\{ F({\mathcal {P}}_n \cup \{y_1,y_2\ldots ,y_{m+1}\})\right\} - {\mathbb {E}}\left\{ F({\mathcal {P}}_n \cup \{y_2,\ldots ,y_{m+1}\})\right\} \big |, \end{aligned}$$

we have that $g > c/2$ Lebesgue a.e. on U. Setting $\nu _n(.)$ to be the intensity measure of the point process ${\mathcal {P}}_n$, we have that

$$\begin{aligned} f_*|.| \le \nu _n(.) \le f^* |. |, \end{aligned}$$

where we recall that |.| stands for the Lebesgue measure on ${\mathbb {R}}^d$. Using these bounds for $\nu _n(.)$, we firstly obtain that

$$\begin{aligned} \nu _n^{m+1}(U) \ge nf_*^{m+1}|A|(nw_dr^d)^m. \end{aligned}$$

Secondly, let $\emptyset \ne J \subset \{1,\ldots ,m+1\}$ and for $y = (y_1,\ldots ,y_{m+1}) \in {\mathbb {R}}^{d(m+1)}$, let $y_J$ be the components of y with indices in J. If $y \in U$, then $|y_i - y_j| \le 2r$ for all $1 \le i,j \le m+1$ and so for any $y_J \in {\mathbb {R}}^{d|J|}$, we have that

$$\begin{aligned} \nu ^{m+1-|J|}_n \big ( \{y_{J^c} \in {\mathbb {R}}^{d(m+1-|J|)} : y \in U \} \big ) \le (2^df^*nw_dr^d)^{m+1-|J|}. \end{aligned}$$

Using the above upper and lower bounds in the proof of [23, Theorem 5.3] along with the a.e. lower bound for g on U, (5.2) follows. $\square$

1.2 Appendix 2

We use the notation of Sect. 4.2, and consider a sequence $\{B_n\}$ of subsets of ${\mathbb {R}}^d$ satisfying Conditions (A)–(D) there.

Given such a sequence, let ${\mathfrak {B}}$ ($={\mathfrak {B}}(\{B_n\})$) be the collection of all subsets A of ${\mathbb {R}}^d$ such that $A = B_n + x$ for some $B_n$ in the sequence and some point $x \in {\mathbb {R}}^d$.

For a $A \in {\mathfrak {B}}$, we shall denote by ${{\mathcal {U}}}_{m,A}$ the point process obtained by choosing m points uniformly in A. Then the extended binomial process ${{\mathcal {U}}}_n$ of Sect. 4.2 is equivalent to ${{\mathcal {U}}}_{n,B_n}$ in the current notation.

Theorem 5.2

[33, Theorem 2.1] Let H be a real-valued functional defined for all finite subsets of ${\mathbb {R}}^d$ and satisfying the following four conditions : 

• 1. Translation invariance: $H({{\mathcal {X}}}+ y) = H({{\mathcal {X}}})$ for all finite subsets ${{\mathcal {X}}}$ and $y \in {\mathbb {R}}^d$.

• 2. Strong stabilization: H is called strongly stabilizing if there exist a.s. finite random variables R (called the radius of stabilization for H) and $D_H(\infty )$ such that,  with probability 1, 

  $$\begin{aligned} (D_OH)\big (({\mathcal {P}}\cap B_O(R)) \cup A\big )= D_H(\infty ) \end{aligned}$$

  for all finite $A \subset B_O(R)^c$.

• 3. Uniformly bounded moments:

  $$\begin{aligned} \sup _{A \in {\mathfrak {B}} ; \, 0 \in A} \, \sup _{m \in \left[ {|A|}/{2},{3|A|}/{2}\right] }{\mathbb {E}}\left\{ \left[ (D_OH)({{\mathcal {U}}}_{m,A})\right] ^4\right\} <\infty . \end{aligned}$$

• 4. Polynomial boundedness: There is a constant $b_2$ such that,  for all finite subsets ${{\mathcal {X}}}\subset {\mathbb {R}}^d,$

  $$\begin{aligned} |H({{\mathcal {X}}})| \le b_2\left[ \mathrm{diam}({{\mathcal {X}}}) +| {{\mathcal {X}}}|\right] ^{b_2}. \end{aligned}$$

  (5.3)

Then,  there exist constants $\sigma ^2,\tau ^2$ with $0 \le \tau ^2 \le \sigma ^2,$ such that,  as $n \rightarrow \infty ,$

$$\begin{aligned}&n^{-1}\mathsf {VAR}\left( H({\mathcal {P}}\cap B_n)\right) \rightarrow \sigma ^2, \qquad n^{-1}\mathsf {VAR}\left( H({{\mathcal {U}}}_n)\right) \rightarrow \tau ^2 , \\&\quad n^{-1/2}\left[ H({\mathcal {P}}\cap B_n) - {\mathbb {E}}\left\{ H({\mathcal {P}}\cap B_n)\right\} \right] \ {\Rightarrow }\ N(0,\sigma ^2), \end{aligned}$$

and

$$\begin{aligned} n^{-1/2}\left[ H({{\mathcal {U}}}_n)) - {\mathbb {E}}\left\{ H({{\mathcal {U}}}_n)\right\} \right] \ {\Rightarrow }\ N(0,\tau ^2), \end{aligned}$$

where $\Rightarrow$ denotes convergence in distribution. The constants $\sigma ^2,\tau ^2$ are independent of the choice of $B_n$. If $D_H(\infty )$ is non-degenerate,  then $\tau ^2 > 0$ and also $\sigma ^2 > 0$. Further,  if ${\mathbb {E}}\left\{ D_H(\infty )\right\} \ne 0,$ then $\tau ^2 < \sigma ^2$.

The last statement follows from the remark below [33, Theorem 2.1]. In the Poisson case the strongly stabilizing condition required for the central limit theorem can be replaced by the so-called weak stabilization condition, as in [33, Theorem 3.1]. H is said to be weakly stabilizing if there exists a random variable $D_{\infty }(H)$ such that

$$\begin{aligned} (D_OH)({\mathcal {P}}\cap A_n)\ \mathop {\rightarrow }\limits ^{a.s. }\ D_{\infty }(H), \end{aligned}$$

(5.4)

for any ${\mathfrak {B}}$-valued sequence $A_n$ growing to ${\mathbb {R}}^d$.