Local central limit theorems in stochastic geometry

We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance. We apply this result to various quantities arising in stochastic geometry, including: size of the largest component for percolation on a box; number of components, number of edges, or number of isolated points, for random geometric graphs; covered volume for germ-grain coverage models; number of accepted points for finite-input random sequential adsorption; sum of nearest-neighbour distances for a random sample from a continuous multidimensional distribution.


Introduction
A number of general central limit theorems (CLTs) have been proved recently for quantities arising in stochastic geometry subject to a certain local dependence. See [18,19,20,21,22] for some examples. The present work is concerned with local central limit theorems for such quantities. The local CLT for a binomial (n, p) variable says that for large n with p fixed, its probability mass function minus that of the corresponding normal variable rounded to the nearest integer, is uniformly o(n −1/2 ). The classical local CLT provides similar results for sums of i.i.d. variables with an arbitrary distribution possessing a finite second moment. Here we are concerned with sums of variables with some weak dependence, in the sense that the summands can be thought of as contributions from spatial regions with only local interactions between different regions.
Among the examples for which we obtain local CLTs here are the following. In Section 3 we give local CLTs for the number of clusters in percolation on a large finite lattice box, and for the size of the largest open cluster for supercritical percolation on a large finite box, as the box size becomes large. In Sections 4 and 5 we consider continuum models, starting with random geometric graphs [18] for which we demonstrate local CLTs for the number of copies of a fixed subgraph (for example the number of edges) both in the thermodynamic limit (in which the mean degree is Θ(1)) and in the sparse limit (in which the mean degree vanishes). For the thermodynamic limit we also derive local CLTs for the number of components of a given type (for example the number of isolated points), as an example of a more general local CLT for functionals which have finite range interactions or which are sums of functions determined by nearest neighbours (Theorem 5.1). This also yields local CLTs for quantities associated with a variety of other models, including germ-grain models and random sequential adsorption in the continuum.
We derive these local CLTs using the following idea which has been seen (in somewhat different form) in [8], in [4], and no doubt elsewhere. If the random variable of interest is known to satisfy a CLT, and can be decomposed (with high probability) as the sum of two independent parts, one of which satisfies a local CLT with the same order of variance growth, then one can find a local CLT for the original variable. Theorem 2.1 below formalises this idea. The statement of this result has no geometrical content and it could be of use elsewhere.
In the geometrical context, one can often use the geometrical structure to effect such a decomposition. Loosely speaking, in these examples one can represent a positive proportion of the spatial region under consideration as a union of disjoint boxes or balls, in such a way that with high probability a non-vanishing proportion of the boxes are 'good' in some sense, where the contributions to the variable of interest from a good box, given the configuration outside the box and given that it has the 'good' property, are i.i.d. Then the classical local CLT applies to the total contribution from good boxes, and one can represent the variable of interest as the sum of two independent contributions, one of which (namely the contribution from good boxes) satisfies a local CLT, and then apply Theorem 2.1. This technique is related to a method used by Avram and Bertsimas [1] to find lower bounds on the variance for certain quantities in stochastic geometry, although the examples considered here are mostly different from those considered in [1].
In any case, our results provide extra information on the CLT behaviour for variables for numerous geometrical and multivariate stochastic settings, which have arisen in a variety of applications (see the examples in Section 5).
We say a random variable X is integrable if E |X | < ∞. We say X has a lattice distribution if there exists h > 0 such that (X − a)/h ∈ Z almost surely for some a ∈ R. If X is lattice, then the largest such h is called the span of X , and here denoted h X . If X is non-lattice, then we set h X := 0. If X is degenerate, i.e. if Var[X ] = 0, then we set h X := +∞. As usual with local central limit theorems, we need to distinguish between the lattice and non-lattice cases. For real numbers a ≥ 0, b > 0, we shall write a|b to mean that either b is an integer multiple of a or a = 0. When a = +∞, b < ∞ we shall say by convention that a|b does not hold. Theorem 2.1. Let V, V 1 , V 2 , V 3 , . . . be independent identically distributed random variables. Suppose for each n ∈ N that (Y n , S n , Z n ) is a triple of integrable random variables on the same sample space such that (i) Y n and S n are independent, with S n = n j=1 V j ; (ii) both n −1/2 E |Z n − (Y n + S n )| and n 1/2 P[Z n = Y n + S n ] tend to zero as n → ∞; and (iii) for some σ ∈ [0, ∞), Remarks. The main case to consider is c n = n 1/2 . The more general formulation above is convenient in some applications, e.g., in the proof of Theorem 4.1. Theorem 2.1 is proved in Section 7. Our main interest is in the conclusion (2.2), but (2.3), which comes out for free from the proof, is also of interest.

Percolation
Most of our applications of Theorem 2.1 will be in the continuum, but we start with applications to percolation on the lattice. We consider site percolation with parameter p, where each site (element) of We write lim inf(B n ) for ∪ n≥1 ∩ m≥n B m .
Theorem 3.1. Suppose d ≥ 2 and p ∈ (0, 1). Then there exists σ > 0 such that if (B n ) n≥1 is any sequence of non-empty finite subsets in Z d with vanishing relative boundary and with lim inf(B n ) = Z d , then For Theorems 3.1 and 3.2 are proved in Section 8. Theorem 3.1 is the simplest of our applications of Theorem 2.1 and we give its proof with some extra detail for instructional purposes.

Random geometric graphs
For our results in this section and the next, on continuum stochastic geometry, let X 1 , X 2 , . . . be i.i.d. d-dimensional random vectors with common density f . Assume throughout that f max := sup x∈R d f (x) < ∞, and that f is almost everywhere continuous. Define the induced binomial point processes In the special case where f is the density of the uniform distribution on the unit For locally finite ⊂ R d and r > 0, let ( , r) denote the graph with vertex set and with edges connecting each pair of vertices x, y in with | y − x| ≤ r; here | · | denotes the Euclidean norm though there should not be any difficulty extending our results to other norms. Sometimes ( , r) is called a geometric graph or Gilbert graph.
Let (r n ) n≥1 be a sequence with r n → 0 as n → ∞. Graphs of the type of ( n , r n ) are the subject of the monograph [18]. Among the quantities of interest associated with ( n , r n ) are the number of edges, the number of triangles, and so on; also the number of isolated points, the number of isolated edges, and so on. CLTs for such quantities are given in Chapter 3 of [18] (see the notes therein for other references) for a large class of limiting regimes for r n . Here we give some associated local CLTs.
Let κ ∈ N and let Γ be a fixed connected graph with κ vertices. We follow terminology in [18]. With ∼ denoting graph isomorphism, let G n be the number of κ-subsets of n such that ( , r n ) ∼ Γ (i.e., the number of induced subgraphs of ( n , r n ) that are isomorphic to Γ). Let G * n (denoted J n in [18]) denote the number of components of ( n , r n ) that are isomorphic to Γ. To avoid certain trivialities, assume that Γ is feasible in the sense of [18], i.e. that ( κ , r) is isomorphic to Γ with strictly positive probability for some r > 0. When considering G n , we shall also assume that κ ≥ 2. We shall give local CLTs for G n and G * n . We assume existence of the limit so that ρ could be zero. If ρ > 0 then we are taking the thermodynamic limit.
We also assume that Then (see Theorems 3.12 and 3.13 of [18]) there exists a constant σ = σ( f , Γ, ρ) > 0, given explicitly in terms of f , Γ and ρ in [18], such that We prove here an associated local central limit theorem for the case f ≡ f U .
We prove Theorem 4.1 in Section 9. It should be possible to obtain similar results for G * n , but we shall do so only for the thermodynamic limit with ρ > 0, as an example in the next section. In the next section we shall see that for the case with ρ > 0, it is possible to relax the assumption that f ≡ f U in Theorem 4.1; when ρ = 0, a similar extension to non-uniform densities should be possible, but we content ourselves here with the case f ≡ f U so as to provide one example where the simplicity and the appeal of the approach do not get buried.

General local CLTs in stochastic geometry
In this section we present some general local central limit theorems in stochastic geometry. We shall illustrate these by some examples in the next section.
For our general local CLTs in stochastic geometry, we consider marked point sets in R d . Let be an arbitrary measurable space (the mark space), and let P be a probability distribution on . Given x = (x, t) ∈ R d × and given y ∈ R d , set y + x := ( y + x, t). Given also a ∈ R, set ax = (a x, t). We think of t as a mark attached to the point x ∈ R d that is unaffected by translation or scalar multiplication. Given * ⊂ R d × , y ∈ R d , and a ∈ (0, ∞), let y + a * : Let ω d denote the volume of the d-dimensional unit ball B (1).
Suppose H( * ) is a measurable R-valued function defined for all finite * ⊂ R d × . Suppose H is translation invariant, i.e. H( y + * ) = H( * ) for all y ∈ R d and all * .
Throughout this section we consider the thermodynamic limit; let r n , n ≥ 1 be a sequence of constants such that (4.2) holds with ρ > 0. Define Let the point process n := {X 1 , . . . , X n } in R d be as given in (4.1), with f as in Section 4 (so f max < ∞ and f is Lebesgue-almost everywhere continuous). Define the corresponding marked point process (i.e., point process in R d × ) by * where (T 1 , T 2 , T 3 , . . .) is a sequence of independent -valued random variables with distribution P , independent of everything else. We are interested in local CLTs for H n ( * n ), for general functions H. We give two distinct types of condition on H, either of which is sufficient to obtain a local CLT.
We shall say that H has finite range interactions if there exists a constant τ ∈ (0, ∞) such that H( * ∪ * ) = H( * ) + H( * ) whenever D( * , * ) > τ. (5.2) In many examples it is natural to write H( * ) as a sum. Suppose ξ(x; * ) is a measurable Rvalued function defined for all pairs (x, * ), where * ⊂ R d × is finite and x is an element of * . Suppose ξ is translation invariant, i.e. ξ( y +x; y + * ) = ξ(x; * ) for all y ∈ R d and all x, * . Then ξ induces a translation-invariant functional H (ξ) defined on finite point sets * ⊂ R d × by Given r ∈ (0, ∞) we say ξ has range r if ξ((x, t); * ) = ξ((x, t); * ∩ B * r (x)) for all finite * ⊂ R d × and all (x, t) ∈ * . It is easy to see that if ξ has range r for some (finite) r then H (ξ) has finite range interactions, although not all H with finite range interactions arise in this way.
Let κ ∈ N. Given any set * ⊂ R d × with more than κ elements, and given x = (x, t) ∈ * , set R κ (x; * ) to be the κ-nearest neighbour distance from x to * , i.e. the smallest r ≥ 0 such that * ∩ B * (x; r) has at least κ elements other than x itself. If * has κ or fewer elements, set R κ (x; * ) := ∞.
We give local CLTs for H under two alternative sets of conditions: either (i) when H has finite range interactions, or (ii) when H is induced, according to the definition (5.3), by a functional ξ(x; * ) which depends only on the κ nearest neighbours, for some fixed κ.
Given K > 0 and n ∈ N, define point processes n,K , and n in R d , and point processes * n,K , and * n in R d × , as follows. Let n,K denote the point process consisting of n independent uniform random points U 1,K , . . . , U n,K in B(K), and let n be the point process consisting of n independent points Z 1 , . . . , Z n in R d , each with a d-dimensional standard normal distribution (any other positive continuous density on R d would do just as well). The corresponding marked point processes are defined by * Theorem 5.1. Suppose that either (i) H has finite range interactions and h H( * n ) < ∞ for some n ∈ N, or (ii) for some κ ∈ N, H is induced by a functional ξ(x; * ) which depends only on the κ nearest neighbours, and h H( * n ) < ∞ for some n ∈ N with n > κ. Suppose also that H n ( * n ) and H( * n,K ) are integrable for all n ∈ N and K > 0. Finally suppose that Then σ > 0 and h(H) < ∞, and for any b ∈ (0, ∞), with h(H)|b, we have ). Suppose that either (i) H has finite range interactions and h H( n ) < ∞ for some n ∈ N, or (ii) for some κ ∈ N, H is induced by a functional ξ(x; ) which depends only on the κ nearest neighbours, and h H( n ) < ∞ for some n ∈ N with n > κ. Suppose also that H n ( n ) and H( n,K ) are integrable for all n ∈ N and K > 0. Finally suppose that Then σ > 0 and h(H) < ∞ and for any b ∈ (0, ∞), with h(H)|b, we have In the next three theorems, we impose some extra assumptions besides those of Theorem 5.1. Writing supp( f ) for the support of f , we shall assume that supp( f ) is compact, and that also r n satisfy which implies (4.2) with ρ = 1. We also assume certain polynomial growth bounds; see (5.11), (5.13) and (5.14) below.
Suppose h H( * n ) < ∞ for some n ∈ N, and suppose supp( f ) is compact. Finally, suppose that (5.10) holds. Then there exists σ ∈ (0, ∞) such that (5.5) and (5.9) hold, and h(H) < ∞ and (5.6) holds for all b with h(H)|b. Now we turn to the general case of Condition (i) in Theorem 5.1, where H has finite range interactions but is not induced by a finite range ξ. For this case we shall borrow some concepts from continuum percolation. For λ > 0, let λ denote a homogeneous Poisson point process in R d with intensity λ. Let * λ denote the same Poisson point process with each point given an independent -valued mark with the distribution P .
Let λ c be the critical value for percolation in d dimensions, that is, the supremum of the set of all λ > 0 such that the component of the geometric (Gilbert) graph G( λ ∪{0}, 1) containing the origin is almost surely finite. It is known (see e.g. [18]) that 0 < λ c < ∞ when d ≥ 2 and λ c = ∞ when d = 1.
Now we turn to condition (ii) in Theorem 5.1. Following [24], we say that a closed region A ⊂ R d is a d-dimensional C 1 submanifold-with-boundary of R d if it has a differentiable boundary in the following sense: for every x in the boundary ∂ A of A, there is an open U ⊂ R d , and a continuously differentiable injection g from U to R d , such that 0 ∈ U and g(0 Theorem 5.4. Let κ ∈ N. Suppose H = H (ξ) is induced by a ξ which depends only on the κ nearest neighbours, and for some γ ∈ (0, ∞) suppose we have for all (x, * ) that Suppose also that supp( f ) is either a compact convex region in R d or a compact d-dimensional submanifold-with-boundary of R d , and suppose f is bounded away from zero on supp( f ). Finally suppose that the sequence (r n ) n≥1 satisfies (5.10), and that h H( * n ) < ∞ for some n ∈ N with n > κ. Then there exists σ ∈ (0, ∞) such that (5.5) and (5.9) hold, and h(H) < ∞ and if b ∈ (0, ∞) with h(H)|b then (5.6) also holds.
We prove Theorems 5.2, 5.3 and 5.4 in Section 11. In proving each of these results, we apply Theorem 5.1, and check the CLT condition (5.5) using a general CLT from [20], stated below as Theorem 11.1.
The conclusion that σ > 0 in Theorems 5.1-5.4 and Corollary 5.1 is noteworthy because the result from [20] on its own does not guarantee this. Our approach to showing σ > 0 here is related to that given in [1] (and elsewhere) but is more generic. A different approach to providing generic variance lower bounds was used in [21] and [3] but is less well suited to the present setting.

Applications
This section contains discussion of some examples of concrete models in stochastic geometry, to which the general local central limit theorems presented in Section 5 are applicable. Further examples where the conditions for these general theorems can be verified are discussed in [20,21,22,23].

Further quantities associated with random geometric graphs
Suppose the graph ( n , r n ) is as in Section 4. We assume here that (4.2) holds with ρ > 0. Theorem 5.1 enables us to extend the case ρ > 0 of Theorem 4.1 to non-uniform f . It also yields local CLTs for some graph quantities not covered by Theorem 4.1; we now give some examples.
Number of components for ( n , r n ). This quantity can be written in the form H n ( n ), where H( ) is the number of components of the geometric graph ( , 1) (which clearly has finite range interactions). In the the thermodynamic limit, this quantity satisfies the CLT (5.7) (see Theorem 13.26 of [18]). Therefore, Corollary 5.1 is applicable here and shows that it satisfies the local CLT (5.8).
Number of components for ( n , r n ) isomorphic to a given feasible graph Γ. This quantity, denoted G * n in Section 4, can be written in the form H n ( n ), with H( ) the number of components of G( , 1) isomorphic to Γ. Clearly, this H has finite range interactions since (5.2) holds for τ = 2. Also, it satisfies (5.7) by Theorem 3.14 of [18]. Therefore we can apply Corollary 5.1 to deduce (5.8) in this case.
Independence number. The independence number of a finite graph is the maximal number k such that there exists a set of k vertices in the graph such that none of them are adjacent. Clearly this quantity is the sum of the independence numbers of the graph's components, and therefore if for ⊂ R d we set H( ) to be the independence number of ( , τ) (also known as the off-line packing number since it is the maximum number of balls of radius τ/2 that can be packed centred at points of ) then H satisfies the finite range interactions condition (5.2) with r = 2. Therefore we can apply Theorem 5.3 to derive a local CLT for the independence number of ( n , r n ), as follows. Theorem 6.1. Let τ > 0 and suppose (5.12) holds. Suppose r n satisfy (5.10). If for ⊂ R d we set H( ) to be the independence number of ( , τ), then there exists σ ∈ (0, ∞) such that (5.7) holds, and if b ∈ N then (5.8) holds.

Germ-grain models
Consider a coverage process in which each point X i has an associated mark T i , the T i (defined for i ≥ 1) being i.i.d. nonnegative random variables with a distribution having bounded support (i.e., with P[T i ≤ K] = 1 for some finite K). Define the random coverage process For U a finite union of convex sets in R d , let |U| denote the volume of U (i.e. its Lebesgue measure) and let |∂ U| denote the surface area of U (i.e. the (d − 1)-dimensional Hausdorff measure of its boundary).

Theorem 6.2.
Under the above assumptions, if (5.10) holds then there exists σ > 0 andσ > 0 such , and moreover for any b ∈ (0, ∞), and given by the volume of that part of the ball centred at x with radius given by the associated mark t, which is not covered by any corresponding ball for some other point x ∈ with x preceding x in the lexicographic ordering. Since we assume the support of the distribution of the T i is bounded, this ξ has finite range r = 2K. Moreover, it satisfies the polynomial growth bound (5.11) so by Theorem 5.2 we get the CLT (5.5) and local CLT (5.6) for any b > 0 (in this example h(H) = 0). Thus we have (6.2).
Turning to the surface area |∂ Ξ n |, this can also be viewed as a functional H n ( n ) for a different H = H (ξ) , this time taking ξ(x; ) to be the uncovered surface area of the ball at x, which again has range r = 2K and satisfies (5.11). Hence by Theorem 5.2. we get the CLT (5.5) and local CLT (5.6) for any b > 0 for this choice of H (in this example, again h(H) = 0). Thus we have (6.3).
Remark. The preceding argument still works if the independent balls of random radius in the preceding discussion are replaced by independent copies of a random compact shape that is almost surely contained in the ball B(K) for some K (cf. Section 6.1 of [20]).
Other functionals for the germ-grain model. When f ≡ f U , the scaled point process r −1/d n n can be viewed as a uniform point process in a window of side r −1/d n . CLTs for a large class of other functionals on germ-grain models in such a window are considered in [13], for the Poissonized point process with a Poisson distributed number of points. Since the Poissonized version of Theorems 5.1 and 5.2 should also hold, it should be possible to derive local CLTs for many of the quantities considered in [13], at least in the case where the grains (i.e., the balls or other shapes attached to the random points) are of uniformly bounded diameter.

Random sequential adsorption (RSA).
RSA (on-line packing) is a model of irreversible deposition of particles onto an initially empty ddimensional surface where particles of fixed finite size arrive sequentially at random locations in an initially empty region A of a d-dimensional space (typically d = 1 or d = 2), and each successive particle is accepted if it does not overlap any previously accepted particle. The region A is taken to be compact and convex. The locations of successive particles are independent and governed by some density f on A. In the present setting, we take the mark space to be [0, 1] with P the uniform distribution. Each point x = (x, t) of * represents an incoming particle with arrival time t. The marks determine the order in which particles arrive, and two particles at x = (x, t) and y = ( y, u) are said to overlap if |x − y| ≤ 1. Let H( * ) denote the number of accepted particles. This choice of H clearly has finite range interactions ((5.2) holds for τ = 2).
Then H n ( * n ) represents the number of accepted particles for the re-scaled marked point process r −1 n * n ; note that the density f and hence the region A on which the particles are deposited, does not vary with n. At least for r n = n −1/d , the central limit theorem for H n ( n ) is known to hold; see [22] for the case when A = [0, 1] d and f ≡ f U and [3] for the extension to the non-uniform case on arbitrary compact convex A (note that these results do not require the sub-criticality condition (5.12) to be satisfied). Thus, the H under consideration here satisfies the condition (5.5). Therefore we can apply Theorem 5.1 to obtain a local CLT for the number of accepted particles in this model. Theorem 6.3. Suppose f has compact convex support and is bounded away from zero and infinity on its support. Suppose r n = n −1/d , and suppose Z n = H n ( * n ) is the number of accepted particles in the rescaled RSA model described above. In other words, suppose Z n be the number of accepted particles when RSA is performed on n with distance parameter r n = n −1/d . Then there is a constant σ ∈ (0, ∞) such that (2.1) holds and for b = 1 and c = n 1/2 , (2.2) holds.
It is likely that in the preceding result the condition r n = n −1/d can be relaxed to (4.2) holding with ρ > 0. We have not checked the details.
In the infinite input version of RSA with range of interaction r, particles continue to arrive until the region A is saturated, and the total number of accepted particles is a random variable with its distribution determined by r. A central limit theorem for the (random) total number of accepted particles (in the limit r → 0) is known to hold, at least for f ≡ f U ; see [25]. It would be interesting to know if a corresponding local central limit theorem holds here as well.

Nearest neighbour functionals
Many functionals have arisen in the applied literature which can be expressed as sums of functionals of κ-nearest neighbours, for such problems as multidimensional goodness-of-fit tests [5,2], multidimensional two-sample tests [14], entropy estimation of probability distributions [17], dimension estimation [16], and nonparametric regression [10]. Functionals considered include: sums of powerweighted nearest neighbour distances, sums of logarithmic functions of the nearest-neighbour distances, number of nearest-neighbours from the same sample in a two-sample problem, and others. Central limit theorems have been obtained explicitly for some of these examples [5,14,2] and in other cases they can often be derived from more general results [1,20,21,7]. Thus, for many of these examples it should be possible to check the conditions of Theorem 5.1 (case (ii)).
We consider just one simple example where Theorem 5.4 is applicable. Suppose for some fixed α > 0 that H( ) is the sum of the α-power-weighted nearest neighbour distances in (for α = 1 this is known as the total length of the directed nearest neighbour graph on ). That is, suppose and ξ clearly satisfies (5.14) for some γ, so provided f is supported by a compact convex region in R d or by a compact d-dimensional submanifold-with-boundary of R d , and provided f is bounded away from zero on its support, Theorem 5.4 is applicable with κ = 1. Hence in this case there exists σ ∈ (0, ∞) such that (5.5) and (for any b ∈ (0, ∞)) (5.6) are valid.

Proof of Theorem 2.1
In the case σ V = 0, Theorem 2.1 is trivial, so from now on in this section, we assume σ V > 0. Let b, c 1 , c 2 , c 3 , . . . be positive constants with h V |b and c n ∼ n 1/2 as n → ∞.
We prove Theorem 2.1 first in the special case where Z n = S n , then in the case where Z n = Y n + S n , and then in full generality. Before starting we recall a fact about characteristic functions.
Proof. See for example Section 3, and in particular the final display, of [26].
Proof. First consider the special case with c n = n 1/2 . In this case, (7.1) holds by the classical local central limit theorem for sums of i.i.d. non-lattice variables with finite second moment in the case where h V = 0 (see page 232 of [6], or Theorem 2.5.4 of [9]), and by the local central limit theorem for sums of i.i.d. lattice variables in the case where h V > 0 and b/h V ∈ Z (see Theorem XV.5.3 of [11], or Theorem 2.5.2 of [9]).
To extend this to the general case with c n ∼ n 1/2 , observe first that by the special case considered above, n 1/2 P[S n ∈ [u, u + b)] remains bounded uniformly in u and n, and hence Also, for any K > 1, Also, for large enough n, and since K is arbitrarily large, combined with (7.3), this shows that Combined with (7.2), this shows that we can deduce (7.1) for general c n satisfying c n ∼ n 1/2 from the special case with c n = n 1/2 which was established earlier. Proof. Assume, along with the hypotheses of Theorem 2.1, that Z n = Y n + S n . Considering characteristic functions, by If σ V = ∞ then by Lemma 7.1, the second factor in the left hand side of (7.4) tends to zero, giving a contradiction. Hence we may assume σ V < ∞ from now on.
By the Central Limit Theorem, By (7.4) and (7.5), σ 2 V ≤ σ 2 and setting σ 2 Let u ∈ R and set Since we assume that Z n = Y n + S n , by independence of Y n and S n we have Hence, Suppose this fails. Then there is a strictly increasing sequence of natural numbers (n(m), m ≥ 1) and a sequence of real numbers (u m , m ≥ 1) such that with t m := t(u m , n(m)), we have By taking a subsequence if necessary, we may assume without loss of generality, either that t m → t for some t ∈ R, or that |t m | → ∞ as m → ∞. Consider first the latter case. If |t m | → ∞ as m → ∞, then by (7.6), ) is equal to t m by (7.7), we also have under this assumption that c n(m) σ tends to zero, and thus we obtain a contradiction of (7.10).
In the case where t m → t for some finite t, we have by (7.6 If we assume W 1 , W 2 are independent, then E f W 2 (t − W 1 ) is the convolution formula for the probability density function of W 1 + W 2 , which is (0, σ 2 ), so that On the other hand, since c −1 n(m) (u m − E Z n(m) ) is equal (by (7.7)) to t m which we assume converges to t, we also have that and therefore we obtain a contradiction of (7.10) in this case too. Thus (7.10) fails, and therefore (7.9) holds. Hence, (2.2) holds in the case with Z n = Y n + S n .
Proof of Theorem 2.1. Set Z n := Y n + S n . By the integrability assumptions, Z n is integrable. By (2.1) and the assumption that n −1/2 E |Z n − Z n | → 0 as n → ∞, Hence, by the assumption n 1/2 P[Z n = Z n ] → 0, and since the assumption n −1/2 E |Z n − Z n | → 0 implies that c −1 n (E Z n − E Z n ) → 0 as n → ∞, and φ is uniformly continuous on R, we can then deduce (2.2).

Lemma 8.1. If X is a binomial or Poisson distributed random variable with
Proof. See e.g. Lemmas 1.1 and 1.2 of [18].
Proof of Theorem 3.1. Let (B n ) n≥1 be a sequence of non-empty finite subsets in Z d with vanishing relative boundary. The first conclusion (3.2) follows from Theorem 3.1 of [19], so it remains to prove (3.3).  Thus S n has the Bin(b n , p) distribution and its distribution, given Y n , is unaffected by the value of Y n so S n is independent of Y n . Also, and |B n | −1/2 E |Λ(B n ) − (Y n + S n )| tend to zero as n → ∞. Combined with (3.2) this shows that Theorem 2.1 is applicable, with h V = 1, and that result shows that (3.3) holds.
In the proof of Theorem 3.2, and again later on, we shall use the following. Lemma 8.2. Suppose ξ 1 , . . . , ξ m are independent identically distributed random elements of some measurable space (E, ). Suppose m ∈ N and ψ : E m → R is measurable and suppose for some finite K that for j = 1, . . . , m, Set Y = ψ(ξ 1 , . . . , ξ m ). Then for any t > 0, Proof. The argument is similar to e.g. the proof of Theorem 3.15 of [18]; we include it for completeness. For 1 ≤ i ≤ m let i be the σ-algebra generated by ξ 1 , . . . , ξ i , and let 0 be the trivial σ-algebra.
, the ith martingale difference. Then with ξ i independent of ξ 1 , . . . , ξ m with the same distribution as them, we have so that |D i | ≤ K almost surely and hence by Azuma's inequality (see e.g. [18]) we have the result.   J(n, 1), . . . , J(n, N n ), with N n := 5 −d |B n |/2 j=1 I n, j . Then we have for n large that Changing the open/closed status of a single site z in B n can change the value of I n, j only for those j for which x n, j − z ∞ ≤ γ n , and the number of such j is at most (2γ n + 1) d . Moreover, for n large so that the total change in N n due to changing the status of a single site z is at most 3 d |B n | 1/4 . So by Lemma 8.2, Then S n has the Bin(m(n), p) distribution and we assert that its distribution, given Y n , is unaffected by the value of Y n so S n is independent of Y n . Indeed, Y n is obtained without sampling the status of the sites x n, j for the first min(m(n), N n ) values of j for which I n, j = 1.
To go into more detail, consider algorithmically sampling the open/closed status of sites in B n as follows. First sample the status of sites outside ∪ j {x n, j }. Then sample the status of those x n, j for which the ∞ -neighbouring sites are not all open (for these sites, I n, j must be zero). At this stage, it remains to sample the status of sites x n, j for which the ∞ -neighbouring sites are all open, and for these sites one can tell, without revealing the value of x n, j , whether or not I n, j = 1 (and in particular one can determine the value of N n ). At the next step sample the status of all x n,i except for the first min(N n , m(n)) values of i which have I n, j = 1. At this point, the value of Y n is determined. However, the value of S n is determined by the status of the remaining unsampled sites together with some extra Bernoulli variables in the case where N n < m(n), so its distribution is independent of the value of Y n as asserted.
Next, we establish that L(B n ) = Y n + S n with high probability. One way in which this could fail would be if N n < m(n), but we know from (8.4) that this has small probability. Also, we claim that with high probability, all sites x n, j for which I n, By Theorem 3.2 of [19], the first conclusion (3.5) holds, and by the preceding discussion, we can then apply Theorem 2.1 with h V = 1, to derive the second conclusion (3.6).

Proof of Theorem 4.1
We are now in the setting of Section 4. Assume f ≡ f U , and fix a feasible connected graph Γ with κ vertices (2 ≤ κ < ∞). Assume also that the sequence (r n ) n≥1 is given and satisfies (4.2) and (4.3).
Then P[ ( κ , 1/(κ + 3)) ∼ Γ] ∈ (0, 1). Let Q n,1 , Q n,2 , . . . , Q n,m(n) be disjoint cubes of side (κ + 5)r n , contained in the unit cube, with m(n) ∼ ((κ + 5)r n ) −d as n → ∞. For 1 ≤ j ≤ m(n), let I n, j be the indicator of the event that n ∩ Q n, j consists of exactly κ points, all of them at a Euclidean distance greater than r n from the boundary of Q n, j . List the indices j ≤ m(n) such that I n, j = 1, in increasing order, as J n,1 , . . . , J n,N n , with N n := m(n) j=1 I n, j . Then E N n = m(n)((κ + 3)/(κ + 5)) dκ P[Bin(n, ((κ + 5)r n ) d ) = κ], (9.1) and hence as n → ∞, since nr d n is bounded by our assumption (4.2), Recalling from (4.3) that τ n := n(nr d n ) κ−1 , we can rewrite (9.2) as as n → ∞. Moreover, for the Poissonized version of this model where the number of points is Poisson distributed with mean n, we have the same asymptotics for the quantity corresponding to N n (the binomial probability in (9.1) is asymptotic to the corresponding Poisson probability). Set α to be one-quarter of the coefficient of τ 2 n in (9.3), if the exponential factor is replaced by its smallest value in the sequence, i.e. set Let N n be defined in the same manner as N n but in terms of n(1−δ) rather than n . That is, set I n, j with I n, j denoting the indicator of the event that n(1−δ) ∩ Q n, j consists of exactly κ points, all at distance greater than r n from the boundary of Q n, j . List the indices j ≤ M n such that I n, j = 1 as J n,1 , . . . , J n,N n .
Since (9.3) holds in the Poisson setting too, using the definition of τ n we have as n → ∞ that By (9.3) and (9.5), we can and do choose δ > 0 to be small enough so that E N n > (3/4)E N n for large n.
By (9.3) and (9.4) we have for large n that 2ατ 2 n ≤ (5/8)E N n . Also, N n is binomially distributed, and hence by Lemma 8.1, P[N n < 2ατ 2 n ] decays exponentially in τ 2 n . By Lemma 8.1, except on an event of probability decaying exponentially in n, the value of M n lies between n(1 − 2δ) and n. If this happens, the discrepancy between N n and N n is due to the addition of at most an extra 2δn points to n (1−δ) . If also N n ≥ 2ατ 2 n then to have N n < ατ 2 n , at least ατ 2 n of the added points must land in the union of the first 2ατ 2 n cubes contributing to N n .
To spell out the preceding argument in more detail, let 1 ≤ j ≤ m(n). If M n < n and I n, j = 1 and X k / ∈ Q n, j for M n < k ≤ n, then I n, j = 1, since in this case n ∩ Q n, j = n(1−δ) ∩ Q n, j . Therefore if M n < n and N n ≥ 2ατ 2 n and n k=M n +1 1{X k ∈ ∪ 2ατ 2 n j=1 Q n,J n, j } < ατ 2 n , then n ∩ Q n,J n, j = n(1−δ) ∩ Q n,J n, j for at most ατ n values of j ∈ [1, 2ατ 2 n ], and hence Hence, if n(1 − 2δ) < M n < n and N n ≥ 2ατ 2 n and M n + 2δn Since nr d n is assumed bounded, we can choose δ small enough so that the expectation of the binomial variable in the last line is less than (α/2)τ 2 n , and then appeal once more to Lemma 8.1 to see that the above conditional probability decays exponentially in τ 2 n . Combining all these probability estimates give the desired result. is taken to be zero. For each j, given that I n, j = 1, the distribution of the contribution to G n from points in Q n, j is Bernoulli with parameter P[ ((κ+3)r n κ , r n ) ∼ Γ], which is p. Hence S n is binomial Bin( ατ 2 n , p). Moreover, the conditional distribution of S n , given the value of Y n , does not depend on the value of Y n , and therefore S n is independent of Y n . By (4.5), Moreover, so that by Lemma 9.1, both τ n P[G n = Y n + S n ] and τ −1 n E |G n − Y n − S n | tend to zero as n → ∞. Hence, Theorem 2.1 (with h V = 1) is applicable, with ατ 2 n playing the role of n in that result and α 1/2 τ n playing the role of c n , yielding as n → ∞. Multiplying through by α −1/2 yields (4.6).

Proof of Theorem 5.1
Recall the definition of h X (the span of X ) from Section 2.
Lemma 10.1. If X and Y are independent random variables then h X +Y |h X .
Proof. If h X +Y = 0 there is nothing to prove. Otherwise, set h = h X +Y . Then, considering characteristic functions, observe that We shall show in both cases (i) and (ii) that h n tends to a finite limit; that is, for both cases we shall show that Then for all > 0 we can pick j ≥ κ + 1 with h j ≤ h + , and then by (10.8) we have h ≤ h + for ≥ j + κ + 1. This demonstrates (10.5) for this case (with h(H) = h ), since we assume h n < ∞ for some n. Moreover, if h(H) > 0, then in the argument just given we can take < h(H) and then for ≥ j + κ + 1 we must have h |h j , which can happen only if h = h j , so by (10.5), in fact h = h j = h(H). That is, we also have (10.6) for this case.
Since we are in the setting of Section 5, we assume (as in Section 4) that f is an almost everywhere continuous probability density function on R d with f max < ∞. The point process n ⊂ R d is a sample from this density, and the marked point process * n ⊂ R d × is obtained by giving each point of n a P -distributed mark. Recall also that we are given a sequence (r n ) with ρ := lim n→∞ nr d n ∈ (0, ∞). Recall from (5.1) that H n ( * ) := H(r −1 n * ) for a given translation-invariant

H.
Our strategy for proving Theorem 5.1 goes as follows. First we choose µ, K as in Lemma 10.2. Then we choose constants β ≥ K and m ≥ µ in a certain way (see below), and use the continuity of f to pick Θ(n) disjoint deterministic balls of radius β r n such that f is positive and almost constant on each of these balls. We use a form of rejection sampling to make the density of points of n in each (unrejected) ball uniform. We also reject all balls which do not contain exactly m points of n in a certain 'good' configuration (of non-vanishing probability). The definition of 'good' is chosen in such a way that the contribution to H n from inside an inner ball of radius K r n is shielded from everything outside the outer ball of radius β r n . We end up with Θ(n) (in probability) unrejected balls, and the contributions to H n ( * n ) from the corresponding inner balls are independent (because of the shielding) and identically distributed (because of the uniformly distributed points) so the sum contribution of these inner balls can play the role of S n in Theorem 2.1.
In the proof of Theorem 5.1, we need to consider certain functions, sets and sequences, defined for β > 0. For x ∈ R d with f (x) > 0, define the function and for x ∈ R d with f (x) > 0 and g n,β (x) > 0, and z ∈ B(x; β r n ), define Since we assume f is almost everywhere continuous, the function g n,β converges almost everywhere on {x : f (x) > 0} to 1. By Egorov's theorem (see e.g. [9]), given β > 0 there is a set A β with is bounded away from zero on A β and g n,β (x) → 1 uniformly on Since we assume (4.2) with ρ > 0 here, for n large enough nr d n < 2ρ. Set Given β > 0, we claim that for n large enough so that nr d n < 2ρ, we can (and do) choose points x β,n,1 , . . . , x β,n, η(β)n in A β with |x β,n, j − x β,n,k | > 2β r n for 1 ≤ j < k ≤ η(β)n . To see this we use a measure-theoretic version of the pigeonhole principle, as follows. Suppose inductively that we have chosen x β,n,1 , . . . , x β,n,k , with k < η(β)n . Then let x β,n,k+1 be the first point, according to the lexicographic ordering, in the set A β \ ∪ k j=1 B(x β,n, j ; 2β r n ). This is possible, because this set is non-empty, because by subadditivity of measure, justifying the claim. Define the ball The balls B β,n,1 , . . . , B β,n, η(β)n are disjoint.
Let W 1 , W 2 , W 3 , . . . be uniformly distributed random variables in [0, 1], independent of each other and of (X j ) n j=1 , where X j = (X j , T j ). For k ∈ N, think of W k as an extra mark attached to the point X k . This is used in the rejection sampling procedure. Given β, if X k ∈ B β,n, j , let us say that the point X k is β-red if the associated mark W k is less than p n,β (x β,n, j , X k ). Given that X k lies in B β,n, j and is β-red, the conditional distribution of X k is uniform over B β,n, j . Now let m ∈ N, and suppose is a measurable set of configurations of m points in B(β) such that P[ m,β ∈ ] > 0. The number m and the set will be chosen so that given there are m points of n in ball B β,n, j , and given their rescaled configuration of lies in the set , there is a subset of these m points which are 'shielded' from the rest of n .

Given
(and by implication β and m), for 1 ≤ j ≤ η(β)n , let I ,n, j be the indicator of the event that the following conditions hold: • The point set n ∩ B β,n, j consists of m points, all of them β-red; • The configuration r −1 n (−x β,n, j + ( n ∩ B β,n, j )) is in .
Let N ,n := η(β)n j=1 I ,n, j , and list the i for which I ,n, j = 1 in increasing order as J( , n, 1) . . . , J( , n, N ,n ). is that of N ,n independent copies of m,β each conditioned to be in .
Proof. Consider first the asymptotics for E [N ,n ]. Given a finite point set ⊂ R d and a set B ⊂ R d , let (B) denote the number of points of in B. Fix m. Since f is bounded away from zero and infinity on A β and g n,β → 1 uniformly on A β , we have uniformly over Hence by binomial approximation to Poisson, and this convergence is also uniform over x ∈ A β .
Given m points X k in B β,n, j , the probability that these are all β-red is at least g n,β (x) m so exceeds 1 2 if n is large enough, since g n,β → 1 uniformly on A β .
Given that m of the points X k lie in B β,n, j , and given that they are all β-red, their spatial locations are independently uniformly distributed over B β,n, j ; hence the conditional probability that r −1 n (−x β,n, j + ( n ∩ B β,n, j )) lies in is a strictly positive constant.
These arguments show that lim inf n→∞ n −1 E [N ,n ] > 0. They also demonstrate part (ii) in the statement of the lemma.
Take δ > 0 with 2δ < lim inf n→∞ n −1 E [N ,n ]. We shall show that P[N ,n < δn] decays exponentially in n, using Lemma 8.2. The variable N ,n is a function of n independent identically distributed triples (marked points) (X k , T k , W k ).

Proof of Theorem 5.1 under condition (i) (finite range interactions). Recall that h(H) is given by (5.4).
Since condition (i) includes the assumption that h H( * n ) < ∞ for some n, by Lemma 10.2 we have h(H) < ∞. Let b > 0 with h(H)|b. Let ∈ (0, b). Let µ ∈ N, and K > 0, be as given by Lemma 10.2.
Choose τ ∈ (0, ∞) such that (5.2) holds. We shall apply Lemma 10.3 with β = K + τ. Let be the set of configurations of µ points in B(K +τ) such that in fact all of the points are in B(K). By Lemma 10.3, we can find δ > 0 such that, writing N n for N ,n we have exponential decay of P[N n < δn].
Let V 1 , V 2 , . . . , be random variables distributed as independent copies of H( * µ,K ), independently of * n . Set Thus, S n is the the total contribution to H n ( * n ) from points in ∪ min( δn ,N n ) =1 B * K+τ,n,J( ,n, ) . By Part (ii) of Lemma 10.3, given that N n ≥ δn, for each we know that r −1 n (−x β,n,J( ,n, ) + * n ) ∩ B * (K + τ) is conditionally distributed as * µ,K+τ conditional on * µ,K+τ ∈ ; in other words, distributed as * µ,K . Therefore the distribution of S n is that of the sum of δn independent copies of H( * µ,K ), independent of the contribution of the other points. Let Y n denote the contribution of the other points, i.e.
Since the distribution of S n , given the value of Y n , does not depend on the value of Y n , S n is independent of Y n .
By assumption H n ( * n ) and S n are integrable. Clearly n 1/2 P[H n ( n ) = Y n + S n ] is at most n 1/2 P[N n < δn], which tends to zero by (10.12). Also by conditioning on N n , we have that which tends to zero by (10.12). This also shows that Y n is integrable By the assumption (5.5), Similarly, setting b 2 := h µ,K b/h µ,K , we have that Since > 0 is arbitrarily small, this gives us (5.6).
Proof of Theorem 5.1 under condition (ii). We now assume that H, instead of having finite range, is given by ( We shall apply Lemma 10.3 with β = 5K, with m = (ν + 1)µ, and with as follows. Let be the set of configurations of m = (ν + 1)µ points in B(β) = B(5K), such that each of 1 , . . . , ν contains at least µ points, and ∪ ν i=1 i contains exactly νµ points, and also the ball 0 contains exactly µ points (so that consequently there are no points in B(5K) \ ∪ ν i=0 i ). A similar construction (using squares rather than balls, and with diagram) was given by Avram and Bertsimas [1] for a related problem.
With this choice of β and , let the locations x β,n, j = x 5K,n, j , the balls B β,n, j = B 5K,n, j , the indicators I ,n, j , and the variables N ,n and J( , n, ) be as described just before Lemma 10.3. By that result, we can (and do) choose δ > 0 such that (10.12) holds. For 1 ≤ ≤ N ,n , the point process r −1 n (−x 5K,n,J( ,n, ) + ( n ∩ B 5K,n,J( ,n, ) )) has µ points within distance K of the origin, and also at least µ points in each of the balls 1 , . . . , ν .
Since µ ≥ κ+1, for any point configuration in , each point inside B(K) has its κ nearest neighbours also inside B(K). Also none of the points in B(5K)\B(K) has any of its κ nearest neighbours in B(K). Also set Y n := H n ( * n ) − S n . Thus S n is the total contribution to H n ( * n ) from points in B * (x 5K,n,J( ,n, ) ; K r n ), 1 ≤ ≤ min( δn , N ,n ). On account of the shielding effect described above, S n is the sum of δn independent copies of a random variable with the distribution of H( * µ,K ). Moreover, we assert that the distribution of S n , given the value of Y n , does not depend on the value of Y n , and therefore S n is independent of Y n .
Essentially, this assertion holds because for any triple of sub-σ-algebras 1 , 2 , 3 , if 1 ∨ 2 is independent of 3 and 1 is independent of 2 then 1 is independent of 2 ∨ 3 (here i ∨ j is the smallest σ-algebra containing both i and j ). In the present instance, to define these σ-algebras we first define the marked point processes j for 1 ≤ j ≤ δn by if N ,n < j ≤ δn . Take   3 to be the σ-algebra generated by the values of J( , n, 1), . . . , J( , n, min( δn , N ,n )) and the locations and marks of points of n outside the union of the balls B 5K,n,J( ,n,1) , . . . , B 5K,n,J( ,n,min( δn ,N ,n )) . Take 2 to be the σ-algebra generated by the point processes j ∩ B * (5K) \ B * (K), 1 ≤ j ≤ δn . Take 1 to be the σ-algebra generated by the point processes j ∩ B * (K), 1 ≤ j ≤ δn . Then by Lemma 10.3 and the definition of , 1 ∨ 2 is independent of 3 and 1 is independent of 2 , so 1 is independent of 2 ∨ 3 . The variable S n is measurable with respect to 1 , and by shielding, the variable Y n is measurable with respect to 2 ∨ 3 , justifying our assertion of independence. By the assumptions of the result being proved, H n ( * n ) and S n are integrable. Clearly n 1/2 P[H n ( * n ) = Y n + S n ] is at most n 1/2 P[N ,n < δn], which tends to zero. Also, as with (10.14) in Case (i), we have that n −1/2 E |H n ( * n ) − (Y n + S n )| tends to zero by (10.12), and Y n is integrable. By (5.5), and so, since h µ,K |b 1 , Theorem 2.1 is applicable with Z n = H n ( * n ), yielding as n → ∞. Multiplying through by δ −1/2 yields (5.6) for this case, when b 1 = b. If b 1 = b, we can complete the proof in the same manner as in the proof for Case (i).

Proof of Theorems 5.2, 5.3 and 5.4
The proofs of Theorems 5.2, 5.3 and 5.4 all rely heavily on Theorem 2.3 of [20] so for convenience we state that result here in the form we shall use it. This requires some further notation, besides the notation we set up earlier in Section 5.
As before, we assume ξ(x, * ) is a translation invariant, measurable R-valued function defined for all pairs (x, * ), where * ⊂ R d × is finite and x is an element of * . We extend the definition of ξ(x, * ) to the case where * ⊂ R d × and x ∈ (R d × ) \ * , by setting ξ(x, * ) to be ξ(x, * ∪ {x}) in this case. Recall that H (ξ) is defined by (5.3).
Let T be an -valued random variable with distribution P , independent of everything else. For λ > 0 let M λ be a Poisson variable with parameter λ, independent of everything else, and let λ be the point process {X 1 , . . . , X M λ }, which is a Poisson point process with intensity λ f (·). Let * λ := {(X 1 , T 1 ), . . . , (X M λ , T M λ )} be the corresponding marked Poisson process. Given λ > 0, we say ξ is λ-homogeneously stabilizing if there is an almost surely finite positive random variable R such that with probability 1, Recall that supp( f ) denotes the support of f . We say that ξ is exponentially stabilizing if for λ ≥ 1 and x ∈ supp( f ) there exists a random variable R x,λ such that for all finite ⊂ (R d \ B(x; λ −1/d R x,λ ))) × , and there exists a finite positive constant C such that For k ∈ N ∪ {0}, let k be the collection of all subsets of supp( f ) with at most k elements. For k ≥ 1 and = {x 1 , . . . , x k } ∈ k \ k−1 , let * be the corresponding marked point set {(x 1 , T 1 ), . . . , (x k , T k )} where T 1 , . . . , T k are independent -valued variables with distribution P , independent of everything else. If ∈ 0 (so = ) let * also be the empty set.
Theorem 11.1 is a special case of Theorem 2.3 of [20], which also provides an expression for σ in terms of integrated two-point correlations; that paper considers random measures given by a sum of contributions from each point, whereas here we just consider the total measure. The sets Ω ∞ and (for all λ ≥ 1) Ω λ in [20] are taken to be supp( f ). Our ξ is translation invariant, and these assumptions lead to some simplification of the notation in [20].
Proof of Theorem 5.2. The condition that ξ(x; * ) has finite range implies that H = H (ξ) has finite range interactions. Since ξ has finite range r, ξ is λ-homogeneously stabilizing for all λ > 0, exponentially stabilizing and binomially exponentially stabilizing (just take R = r, R x,λ = r and R x,λ,n, = r).
We shall establish (5.5) by applying Theorem 11.1. We need to check the moments conditions (11.2) and (11.3) in the present setting. Since we assume that f max < ∞, for any λ > 0 and any n ∈ N with n ≤ 2λ, and any x ∈ supp( f ), the variable card( * n ∩ B * (x; rλ −1/d )) is binomially distributed with mean at most ω d f max 2r d . Hence by Lemma 8.1, there is a constant C, such that whenever n ≤ 2λ and x ∈ supp( f ) we have Moreover by (5.11) and the assumption that ξ has range r, for so by (11.4) we can bound the fourth moments of ξ((λ 1/d x, T ); λ 1/d ( * n ∪ * )) uniformly over (x, λ, n, ) ∈ supp( f ) × [1, ∞) × N × 3 with n ≤ 2λ. This gives us (11.3) (for p = 4 and = 1/2) and (11.2) may be deduced similarly.
Proof of Theorem 5.3. Under condition (5.2), the functional H( * ) can be expressed as a sum of contributions from components of the geometric (Gilbert) graph ( , τ), where := π( * ) is the unmarked point set corresponding to * (recall that π denotes the canonical projection from R d × onto R d .) Hence, H( * ) can be written as H (ξ) ( * ) where ξ(x; * ) denotes the contribution to H( * ) from the component containing π(x), divided by the number of vertices in that component. Then ξ(x; * ) is unaffected by changes to * that do not affect the component of ( , τ) containing π(x), and we shall use this to demonstrate that the conditions of Theorem 11.1 hold, as follows (the argument is similar to that in Section 11.1 of [18]).
Consider first the homogeneous stabilization condition. For λ > 0, let R(λ) be the maximum Euclidean distance from the origin, of vertices in the graph ( λ ∪{0}, τ) that are pathwise connected to the origin. By scaling (see the Mapping theorem in [15]), R(λ) has the same distribution as τ times the maximum Euclidean distance from the origin, of vertices in ( τ d λ ∪ {0}, 1) that are pathwise connected to the origin. Then R(λ) is almost surely finite, for any λ ∈ (0, τ −d λ c ).
Changes to λ at a distance more than R(λ) + τ from the origin do not affect the component of ( λ ∪ {0}, τ) containing the origin and therefore do not affect ξ((0, T ); * λ ). This shows that ξ is λ-homogeneously stabilizing for any λ < τ −d λ c , and therefore by assumption (5.12) the homogeneous stabilization condition of Theorem 11.1 holds.
Next we consider the binomial stabilization condition. Let x ∈ supp( f ). Let R x,λ,n be equal to τ plus the maximum Euclidean distance from λ 1/d x, of vertices in (λ 1/d ( n ∪ {x}), τ) that are pathwise connected to λ 1/d x. Changes to n at a Euclidean distance greater than λ −1/d R x,λ,n from x will have no effect on ξ((λ 1/d x, T ); λ 1/d * n ). Using By scaling, the second probability in (11.5) equals the probability that there is a path from the origin in ( τ d (1+ ) 2 f max ∪ {0}, 1) to a point at Euclidean distance greater than τ −1 u − 1 from the origin. By the exponential decay for subcritical continuum percolation, (see e.g. Lemma 10.2 of [18]), this probability decays exponentially in u (and does not depend on n).
Changes to n ∪ at a Euclidean distance greater than λ −1/d R x,λ,n, from x will have no effect on ξ((λ 1/d x, T ); λ 1/d ( * n ∪ * )); that is, (11.1) holds. To check the tail behaviour of R x,λ,n, , suppose for example that has three elements, x 1 , x 2 and x 3 . Then it is not hard to see that R x,λ,n, ≤ R x,λ,n + R x 1 ,λ,n + R x 2 ,λ,n + R x 3 ,λ,n , and likewise when has fewer than three elements. Using this together with (11.6), it is easy to deduce that there is a constant C such that for all (x, n, , λ, u) ∈ supp( f ) × N × 3 × [1, ∞) 2 with n ≤ (1 + )λ, and we have P[R x,λ,n, > u] ≤ C exp(−u/C). (11.7) In other words, ξ is binomially exponentially stabilizing.
Thus our ξ satisfies all the assumptions of Theorem 11.1, and we can deduce (5.5) and (5.9) for some σ ≥ 0 by applying that result with λ(n) = r −d n . Then by applying Theorem 5.1, we can deduce that σ > 0 and h(H) < ∞ and (5.6) holds whenever h(H)|b.
Proof of Theorem 5.4. Suppose the hypotheses of Theorem 5.4 hold, and assume without loss of generality that ξ(x, * ) = 0 whenever * \ {x} has fewer than κ elements. We assert that under these hypotheses, there exists a constant C such that for all (x, n, λ, u) ∈ supp( f ) × N × [1, ∞) 2 with n ∈ [λ/2, 3λ/2] and n ≥ κ, we have P[λ 1/d R κ ((x, T ); * n ) > u] ≤ C exp(−C −1 u). (11.11) Indeed, if supp( f ) is a compact convex region in R d and f is bounded away from zero on supp( f ), then (11.11) is demonstrated in Section 6.3 of [20], while if supp( f ) is a compact d-dimensional submanifold-with-boundary of R d , and f is bounded away from zero on supp( f ), then (11.11) comes from the proof of Lemma 6.1 of [24].