Correlation of clusters: Partially truncated correlation functions and their decay

In this article, we investigate partially truncated correlation functions (PTCF) of infinite continuous systems of classical point particles with pair interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write the solutions of these equations as a sum of contributions labelled by certain forests graphs, the connected components of which are tree graphs. We generalize the method introduced by R.A. Minlos and S.K. Pogosyan (1977) in the case of truncated correlations. These solutions make it possible to derive strong cluster properties for PTCF which were obtained earlier for lattice spin systems.


Introduction
Correlation functions were first introduced in statistical mechanics by Ornstein and Zernike [26] at the beginning of the 20th century in the study of critical fluctuations. Mathematical studies apparently began with the work of Yvon [38] and the independent researches of Bogolyubov [4], Kirkwood [16] and Born-Green [6], and, in some sense, were completed in the works of Penrose ( [28]), Ruelle (see [35]) and Bogolyubov, Petrina, Khatset [5]. Correlation functions are the probability densities of correlation measures and were called m-particle distribution functions by N. N. Bogolyubov, which more accurately describes their meaning. The physical correlations between particles are in fact described by the socalled truncated correlation functions (TCF) (or connected correlation functions), which become zero in the absence of interaction between the particles.
When studying the thermodynamic properties of statistical systems, the important characteristics are often interactions between groups of particles (so-called clusters). Correlations between clusters are described by the so-called partially truncated correlation functions (PTCF) (or partially connected correlation functions). In the article [20], Lebowitz derived bounds on the decay of correlations between two widely separated sets of particles (two point-PTCF) for ferromagnetic Ising spin systems in terms of the decay of the pair correlation. Later, in [10], some "physically reasonable" hypotheses on the decay of the TCF and PTCF were presented and discussed. In subsequent publications of these authors ( [11,12]) various strong decay properties were proved for TCF of lattice and continuous systems in different situations. And in [14] some general results on strong cluster properties of TCF and PTCF for lattice gases are presented. (In fact, the proof of their main theorem involves long technical parts which were obtained in unpublished work of one of the authors (B.S.).) Here, we consider classical continuous systems of point particles which interact through a two-body interaction potential. We derive for the PTCF equations of Kirkwood-Salsburg type and apply the technique that was proposed in [23] to obtain solutions of these equations in the form of a series of contributions of certain diagram-forests. Such a representation makes it possible to obtain strong cluster properties for the PTCF in a convenient form for deriving estimates. The positions {x i } i∈N of identical particles are assumed to form a locally finite subset in R d . Because the particles are assumed to be identical, the ordering is irrelevant. Moreover, there can be more than one particle at any point. The configuration space is therefore given by locally finite maps: where N 0 = N ∪ {0}. For any Λ ∈ B c (R d ) we denote by γ Λ the restriction of γ to Λ. We also need to define the space of finite configurations Γ 0 in R d : The topology on Γ is generated by the subbasis {O m K } where K runs over compact subsets of R d with nonempty interior, and m ∈ N 0 , given by γ(x) = m}.
Assume that the functions F i : Γ Xi → R (i = 1, 2) are measurable. Then the following identity holds: In the articles [32,30,31] this property is the main technical tool in proving the existence of correlation functions in the infinite-volume limit.
The following identity is similar, and will be used extensively.
Lemma 2.2 For all positive measurable functions G : Γ Λ → R and H : Γ Λ × Γ Λ → R the following identity is true.
For a given j ∈ C ∞ 0 (R d ) with |j| ≤ 1 we define the function e j : Γ 0 → R, by Clearly, e j ∈ D(Γ 0 ). For any η ∈ Γ 0 we define distributions δ η such that for any G ∈ D(Γ 0 ) In terms of ordinary distributions this means that . (2.11) The product in the last line is a direct product of δ-functions. (We put |γ| = x∈R d γ(x).) Note that, if η 1 · η 2 = 0 then δ η1 and δ η2 commute so that for a collection ( (2.12) In distributional form we have taking into account the equation (2.7) in the sense of distributions and the fact that for η i · η j = 0 (2.14) 3 Correlation functions
Remark 3.1 The conditions (3.1) and (3.2) are more restrictive than needed to obtain the basic expansions for the correlation functions. Sufficient assumptions for obtaining analytic expansions are stability:

Gibbs measure
With the notation above, the Gibbs measure in finite volume Λ has the form: A survey and discussion of problems related to the construction of limit Gibbs measures for infinite systems in the space Γ can be found, for example, in the review [19].

Correlation measure and correlation functions
Correlation functions are the analogue of the moments of a measure. Consider the moments of a measure in the configuration space Γ. With every configuration γ ∈ Γ can be associated an occupation measure (see, e.g., [1], [15] where δ x is the Dirac measure, i.e. and C 0 (R d ) is the space of continuous functions with compact support and M + (R d ) is the space of nonnegative Radon measures in B(R d ).
To generalise this to the case of several variables, note that the product of distributions is not defined. For example, in the case of Gaussian measures, one usually applies Wick regularization (see [37,3]). An analogous procedure is used for Poisson variables.
Let H : Γ 0 → R be a function on the configuration space Γ 0 such that H Γ (n) = H (n) ({x 1 , ..., x n }) = H n (x 1 , ..., x n ), (3.13) where H n ∈ C 0 (R dn ) is a symmetric function. Then and we define the n-th power by The correlation measures ρ (n) are defined by In case the correlation measure ρ (n) is absolutely continuous with respect to Lebesgue measure in R dn , correlation functions are defined by Summing equation (3.19) over n ≥ 0, we can now define the correlation measure ρ on Γ Λ for Λ ∈ B c (R d ) by (3.20) and in the case that the correlation measures are absolutely continuous, Using formula (3.21) for the case of the space Γ Λ together with equation (3.9) and formula (2.7), it is easy to obtain the following expression for the finite-volume correlation functions The question of constructing correlation functions in the infinite-volume limit was discussed in detail in [35,5,36,32,30,31].

Truncated (connected) correlation functions
Correlations between particles are better described by truncated (connected) correlation functions. These functions are defined recursively by ρ T (x 1 ) = ρ({x 1 }) and where n = |η| and the asterisk over the sum means that the sum is over all partitions of the set {1, . . . , n} into k non-empty disjoint subsets, and η I = γ x I . That is, They can be also be written explicitly in terms of the correlation functions ρ(η) as follows. Clearly, ρ T (x 1 , . . . , x n ) is permutation-invariant and can be written as ρ T (γ (x1,...,xn) . In case ρ(η) is given by (3.22), the functions ρ T (η) have the following representation in terms of integrals with respect to the measure λ zσ . the following representation is true, Here the function Φ T (γ) is the Ursell function (see, e.g., [35]) defined by for |γ| = 1, {x,y}∈L(G) C xy , for |γ| ≥ 2, (3.28) in which G T (γ) is the set of all connected graphs G (Mayer graphs) with vertices in the points x of the configuration γ, and L(G) is the set of all lines of the graph G, and For the proof see [28], [34] (see also [29] and [35], Chapter 4). In his proof, Penrose noted that one could associate with each connected graph G on γ a unique Cayley tree obtained by deleting bonds from G in a particular way (tree graph identity). The sum over connected graphs may be rearranged by grouping together all terms (graph contributions) corresponding to a given Cayley tree, which are obtained by the procedure of "deleting". Later, Brydges and Federbush [8] invented a new method of deriving the Mayer series for the pressure via a new type of tree graph identity. A more detailed history of the subject and some new results can be found in [25].
In this article we derive an expansion for more general PTCF using the technique of Minlos and Pogosyan [23] which is related to Penrose's original proof ( [28]). A representation for the functions ρ T (x 1 , ..., x m ) in the form of expansions in terms of contributions from tree graphs follows as a special case.

Partially truncated (connected) correlation functions
Partially truncated (connected) correlation functions (PTCF) describe correlations between clusters of particles. Decay estimates for these correlations are an important technical tool in the proof of many physical hypotheses (see, for example, the estimate (4.2) in [7]).
Suppose that η = m i=1 η i . Then PTCF corresponding to this decomposition are defined recursively by: (3.30) where η l = i∈I l η i . Obviously, this definition coincides with (3.23) when all configurations η i consist of one point. Clearly, (3.25), the PTCFρ T m (η) can also be expressed directly in terms of the ρ(η i ): (3.31) (To emphasize the number of clusters we added the index m.) To derive expressions for these functions we define a generating functional. It is a generalization of the generating functional introduced in [14] for spin systems. For a given j ∈ C ∞ 0 (R d ) we define the smoothed correlation function ρ j by Using definitions (2.9) -(2.13), we now put Note, that if j(x) = 1 1 Λ (x) and α 1 = α 2 = · · · = α m = 0 the formula (3.34) is the grand partition function (3.10). Define also We call k-point j-PTCF the following functions: The j-PTCF are given by the following formula, where the second sum in (3.38) runs over all partitions of {1, . . . , m} into r non-empty subsets J 1 , . . . , J r with the restrictions (3.24).
In particular, for j(x) = 1 1 Λ (x) the functions (3.37) are the finite-volume PTCF in Λ, and for j = 1 they are the PTCF in R d .
Proof. This follows from the following formula This formula follows easily by induction. Replacing Z(α 1 , . . . , α m ) by Z j (α, η) m 1 , we have by (2.13), Setting the remaining α i = 0, only the empty set I = ∅ survives and we have (3.42) Taking the limit j → 1, this is (3.31).
Changing the order of summations over indices I and over sets ξ we can write it as This can be rewritten as follows where η I = i∈I η i , and where the asterisk over the second sum means that for all i ∈ I, ξ · η i = 0. (Note that ρ T (η 1 ; . . . ; η m ) = 0 if any of the η i = ∅. ) These equations hold provided η 1 = ∅. They express ρ T m in terms of ρ T k with k ≤ m. Clearly, they therefore determine ρ T m uniquely if the operator

Solution
Due to the assumption that U (γ) = +∞ if γ(x) > 1 for some x, we can restrict ourselves to configurations such that γ ≤ 1. We then adopt set notation and write γ for the set of points x with γ(x) = 1. Following [23] we will now seek a solution of (4.8) in the form: Inserting the expressions for ρ T j;m and ρ T j;m−|I| of the form (4.9) in both sides of (4.8) and applying Lemma 2.2 we arrive at the following recursion relations for the kernels T m , owing to the arbitrariness of the function j, (4.10) Subject to the initial conditions and for m > 1, T m (η 1 ; ...; η m | γ) = 0 if γ = ∅ and any of η i = ∅, (4.12) this equation has a unique solution due to its recursive structure. (These conditions follow from the fact that ρ j (∅) = 1 and ρ T j:m (η 1 ; . . . ; η m ) = 0 if η j = ∅ for some j = 1, . . . , m.) In order to prove that (4.9)-(4.12) is a solution of the equations of (4.8) as j → 1 (in the infinite volume limit), it is necessary to show that the kernels T m (η 1 ; ...; η m | γ) are integrable functionals of the variable γ with respect to the measure λ σ . Using assumption (3.7) we have, from (4.10), (4.13) Following [23], for every number h > 0 and any given bounded positive even function ν : R d → [0, +∞), we consider new kernels Q m (η 1 ; ...; η m | γ), which are uniquely determined by the following system of recursion relations: with initial conditions and The following lemma is true.
Lemma 4.2 Let the parameters β > 0, z > 0 and assume that the interaction potential φ is such that The proof is trivial by induction. The solutions Q m (η 1 ; ...; η m | γ) of the equations (4.14)-(4.17) can be written with the help of forest graphs. For each set of clusters {η 1 ; ...; η m }, η j ∈ Γ 0 \ ∅ and a configuration γ ∈ Γ 0 , we define the set of graphs S η1;...;ηm|γ in the following way. The connected components of the graphs f ∈ S η1;...;ηm|γ are tree graphs with vertices given by points of η 1 ∪ ... ∪ η m ∪ γ, and such that there are no lines (or edges) connecting vertices of the same cluster η i (for i = 1; ..., m). Each tree contains a point of η 1 ∪ · · · ∪ η m and if i 0 is the lowest index such that η i0 contains a point of the tree then this point is unique (the root of the tree). Moreover, for every other vertex z of the tree there is a path z 1 , . . . , z k such that z k = z, and there is an edge between the root x 0 and z 1 and between each pair z p and z p+1 , and such that if z p ∈ η ip then if z p+1 ∈ η ip+1 then z p is the only point in η ip connected to a point in η ip+1 by a line in the forest, whereas if z p+1 ∈ γ then z p is the only point in η ip to which it is connected by a line in the forest. (Note that a single point x ∈ η 1 ∪ · · · ∪ η m is also a tree with analytic contribution h.) Finally, if all points of the configurations η i (for every i = 1, ..., m) are combined into one single vertex, then the forest graph f ∈ S η1;...;ηm|γ reduces to a connected tree graph with m + n vertices, where n = |γ|.
For every forest f ∈ S η1;...;ηm|γ let E( f ) be the set of its edges.
The equation (4.14) reduces to In particular, This agrees with (4.20) since the only allowed tree consists of individual points x ∈ η 1 .
We now do induction on m and l 1 = |η 1 |. For m = 1 we already have that Q 1 (η 1 | ∅) = h l1 . Assuming that Q 1 . . . , Q m−1 are given by the sum of forest contributions when γ = ∅, the terms in (4.23) correspond to the construction of a forest on η 1 ∪ · · · ∪ η m as follows. The point x 1 is connected to a set of points η outside η 1 . If I is the set of indices such that η ∩ η i = ∅, then in Q m−|I| (η 1 \ {x 1 } ∪ η I ; η I c | ∅) there are no more connections within η 1 \ {x 1 } ∪ η I , i.e. between any other point of η 1 and points of ∪ i∈I η i or between two points of ∪ i∈I η i . In Q m−|I| either m − |I| < m or I = ∅, in which case the first subset is η 1 = η 1 \ {x 1 } and |η 1 | < |η 1 |. Therefore, by the induction hypothesis, its contributions are forest graphs with vertices in η 1 ∪ η 2 ∪ · · · ∪ η m such that each tree contains at most one point of η 1 ∪ η I . This means that when the connections with x 1 are added, the resulting graph still consists of separate trees. Denote the resulting forest graph on η 1 ∪ · · · ∪ η m by f . If x = x 1 is a vertex in f then by the induction hypothesis, there is a sequence of points z 0 ∈ η 1 ∪ ∪ m i=2 η i , z 1 , . . . , z k ∈ η I c such that z k = x and if z p ∈ η ip (p = 0, . . . , k) then z p is the unique point in η ip connected to a point in η ip+1 by a line in f .
This is similar to the case γ = ∅. It corresponds to the case where x 1 is only connected to points in η 2 ∪ · · · ∪ η m , and the remaining tree after collapsing the points {x 1 } ∪ η I gives the stated contribution by induction, since either m − |I| < m or |η 1 \ {x 1 }| < |η 1 |. (Once again the contribution of I = ∅ is zero if The other terms are more complicated. Now x 1 is connected to a set of points ξ ⊂ γ as well as a set of points η ⊂ ∪ m i=2 η i . Collecting the points of {x 1 } ∪ ξ ∪ η into a single vertex, the corresponding forest is just the contribution to Q m−|I| (η 1 \ {x 1 } ∪ η I ∪ ξ; η I c | γ \ ξ) by induction since |γ \ ξ| < n. For y ∈ ξ, there are no more lines between other points of η 1 and y. Also, there are no more lines connecting x ∈ η I to another point of η 1 .
It remains to show that upon collapsing the points of each η i to a single vertex, the resulting graph is a connected tree. This is more intricate. We first prove connectedness.
To see that the resulting graph is a tree, note that in any contributing forest to Q m−|I| (η 1 ∪ξ ∪η I ; η I c | γ \ ξ) there is just one line between a point of η 1 ∪ ξ ∪ η I and a tree on η I c ∪ γ \ ξ. The factor K ν (x 1 ; ξ ∪ η) gives lines between x 1 and the points of ξ ∪ η, and therefore to only one point of this tree.
Proof: If y i is an end vertex of a tree in the forest f , the it contributes a factor ν 1 by (4.25). The same holds if from y i outwards there are only vertices y k since we can integrate them successively. In the case when y i lies between the points x i and x k in η 1 ∪ · · · ∪ η m , we first use the inequality ν(y i − x k ) ≤ ν 0 .

Proof:
It is easy to see from relation (4.14) that N  We now prove that, given the initial conditionÑ 1 0 (l 1 ) = 1, N (m) n (l 1 ; . . . ; l m ) = l 1 (l + n) m+n−2 (4.31) is the solution of the recurrent relations (4.30) with l as in (4.22). Inserting in the right-hand side of (4.30) it becomes equal to In the other two sums, this summation is easy, and we obtain and where we used the identity We conclude that which completes the induction, and hence proves (4.31).
Remark 4.6 Formula (4.27) is a generalization of Cayley's well-known formula for the number of tree graphs with n vertices: K n = n n−2 for the case of forest graphs of the system of m clusters η 1 , . . . , η m and n single vertices (l j = |η j |, n = |γ|).
Now we can formulate the theorem of the existence of a solution of the equations (4.8) in the thermodynamic limit j → 1. where |T m (η 1 ; . . . ; η m | γ)| is bounded by a power series in the activity z with integrable coefficients, which converges in the region ze 2βB+2 ν 1 < 1.
where G ν ( f ) is a monomial in z of order l + n. Inserting the bounds (4.26) and (4.27) we get the following estimate Applying Stirling formula n! > n n e −n √ 2πn (see [2], formula 6.1.38), we have: where contribution C xy for an edge of G C ( f ) connecting vertices x and y is given by (3.29) and where the analytic expression for G C ( f ) has the more complicated form where S( f ) is the set of pairs of points of the set η 1 ∪ ... ∪ η m ∪ γ for which there are no edges in the forest f . Consider for example forest diagrams f ∈ S η1;...;ηm|{y1} . Restricting the diagram to η 1 ∪ · · · ∪ η m one obtains a forest on η 1 ∪ · · · ∪ η m of which some trees are connected by an edge in f to y 1 . If there is just one such edge the corresponding contribution is obtained from that of the restricted forest by multiplying by the function ν(x j − y 1 ) if x j is the vertex attached to y 1 . In general one has to multiply by a factor p r=1 ν(x jr − y 1 ). In the former case, integration with respect to the variable y 1 simply multiplies the contribution of the diagram from S η1;...;ηm|∅ by the factor ν 1 . In the general case, we need to consider integrals of the form p r=1 ν(x r − y) dy. Lemma 4.9 Define the kernel ν with polynomial decay by
Proof. We subdivide the integral over y into domains where |y − x r | < k =r |y − x k |. Then |x k − y| > 1 2 |x k − x r | and the inequality (4.45) follows from To count the possible diagrams, we first isolate the parts of the diagram consisting of trees with vertices in γ except possibly one endpoint. This leads to the following expression: see Proposition 4.1.
Proof. This can be proved inductively from the formula (4.46). However, it is also easily understood graphically as follows. Given a graph in S η1;η2;...;ηm|{y1,...,yn} , consider the points of γ = {y 1 , . . . , y n } connected to only one other vertex (endpoints). These are parts of trees on γ with a single base point either in γ or in η 1 ∪ · · · ∪ η m . Starting at the endpoints the corresponding points y i can easily be integrated, yielding factors hν 1 . In the remaining graph, each point of γ is connected to at least two other vertices. We denote the contribution of this graph by Q m (η 1 ; η 2 ; ...; η m | k), where k is the number of remaining vertices in γ. Conversely, given a graph in S η1;η2;...;ηm|{y1,...,y k } in which each point y i (i = 1, . . . , k) is connected to at least two other vertices, we obtain the contribution from graphs in S η1;η2;...;ηm|{y1,...,yn} with n ≥ k, containing this graph and such that all other points y k+1 , . . . , y n are in trees with a single base point, by counting the number of possibilities of attaching trees to the given tree with total number of vertices equal to n − k. But this number is given precisely by Indeed, first we can choose which of the total of n points belongs to the original graph in n k ways and order them in k! ways. The number of ways of forming trees out of the remaining n − k points is then given by N (1) n−k (l+k), because for this purpose we can consider all points of the original graph as belonging to a single cluster as they cannot be connected further to each other. There are obviously l + k such points to be connected to a further n − k external points. According to Lemma 4.5 this can be done in N (1) n−k (l + k) ways.

The case m = 2.
Now consider first the case m = 2 of two clusters. There are two possibilities: either there is at least one line between η 1 and η 2 in the forest, or there is none. In the first case, the restriction of the forest to γ splits into separate trees, each of which is connected to a single point of either η 1 or η 2 . In the second case, the restriction to γ also splits into separate trees, but one of these is connected to a single point of η 1 as well as one or more points of η 2 . The others are again connected to a single point of either η 1 or η 2 . The trees connected to a single point are easily integrated out, giving rise to factors ν 1 . If there is a tree connecting η 1 and η 2 then there is one point y 1 of that tree in γ connected to a point of η 1 and one point y 2 ∈ γ connected to one or more points of η 2 (y 1 can be equal to y 2 ). In that case there is a unique path in the tree connecting y 1 to y 2 . The remaining part of the tree consists of individual trees connected to single points of this path (or points of η 1 ∪ η 2 ). These can be integrated out giving factors ν 1 as before. In terms of the above Proposition 4.1, we have and where K (0) (x 1 ; η 2 ) is given by (4.48) and for k ≥ 1.
We now assume that ν is polynomially bounded, i.e. ν(x) ≤ Cν(x) for some constant C > 0 (and α > d). Integrating over the points on the path from y 1 to y k−1 (including y 1 ) using Lemma 4.9 yields factors 2 1+α Cν 1 . That is, (4.53) The latter integral can be estimated as follows. By Lemma 4.9 we have for any k = 1, . . . , p. Summing over η ⊂ η 2 with x ∈ η, we then have In summing over the trees connected to a single point of this path, the number of vertices in these trees is unlimited. This means that we can consider these trees individually, having base points on the k +l points of the path from η 1 to η 2 and containing n i + 1 points (i = 1, . . . , k +l). There are (n i + 1) ni−1 such trees for each i, so we now have in total, This obviously holds if h (ν 1 e + 2 1+α Cν 1 ) < 1.
(In the first line, there is a factor n! k! n 1 ! . . . n k+l ! for the number of ways of distributing the vertices in γ over the individual trees and the remaining k points of γ and a factor k! for the number of ways of ordering the vertices in the path connecting the two clusters as well as a factor 1/n! from the definition of the correlation function. In the second line, we used the inequality (k + 1) k−1 ≤ k! e k for k ≥ 1.) Remark 4.10 Note that comparing this formula with the expression (4.50), we have the following remarkable identity, where we replaced n − k by n and k + l by l.

The case m = 3.
Here the situation is not too much more complicated. The cases where there is a line between at least one pair of η 1 , η 2 and η 3 reduce to the case m = 2. There remains the case that there is a tree on γ which is connected to all three. Again, this tree has only one point in γ which connects to η i for each i = 1, 2, 3, and by integrating out over intermediate y's which connect to only two others, this reduces to the case where these three points coincide. Assuming that the points connecting the tree to η 1 , η 2 and η 3 are different points y 1 , y 2 and y 3 , there are 3 possible permutations of these points, and we can integrate out any intermediate points as before, yielding factors 2 1+α ν 1 . From Proposition 4.1 we have where Q 3 (η 1 ; η 2 ; η 3 | k) is the contribution from all forest graphs in S η1;η2;η3|{y1,...,y k } , in which all vertices y i of γ are connected to at least two other vertices. Integrating out the vertices of γ connected to only 2 others yields factors 2 1+α ν 1 and results in a tree on γ where every vertex is connected to at least 3 others. There is only one such tree. It consists of a single point y of γ connected to η 1 , η 2 and η 3 . Conversely, given this tree, one can form trees with additional vertices connected to two points by adding a sequence of points between y and η 1 , η 2 , and η 3 . In total, Q 3 (η 1 ; η 2 ; η 3 | k) is the sum of 3 terms: where Q 3,3 (η 1 ; η 2 ; η 3 | k) contains the contributions of terms where there is no connection inside η 1 ∪η 2 ∪η 3 (3 components), Q 3,2 (η 1 ; η 2 ; η 3 | k) corresponds to the terms where there is one or more line(s) between one pair of η 1 , η 2 and η 3 (2 components), and Q 3,1 (η 1 ; η 2 ; η 3 | k) contains the contributions where all 3 clusters are connected by lines inside η 1 ∪ η 2 ∪ η 3 . In the latter, we must have k = 0 since there cannot be another (outside) connection between two η i 's. The corresponding contributions are Moreover, k = 0 for this term in (4.58). The sum over trees is (4.61) Denoting the corresponding decomposition of Q 3 by Q 3 = Q 3,1 + Q 3,2 + Q 3,3 , we have, replacing ν(x − x ) by Cν(x − x ) and noting that * In the cases where only one pair of η 1 , η 2 and η 3 are connected, we have where As in the case m = 2, we assume that ν(x) ≤ Cν(x) where ν is given by (4.44). Then the kernel K (k) is estimated as before: (4.66) Also Therefore, Summing over the external trees, we have There remains the case where there is no line between any points of η 1 ∪ η 2 ∪ η 3 . As explained above, this contribution equals Inserting the bound (4.66), we get Summing over trees attached to points of these paths now gives In conclusion, we can now write where T 3 denotes the trees on three points, and and (4.75) and (4.77)

The case of general m.
As before, we integrate out intermediate points y, which connect to only two others (as well as trees of points y connected to a single point of η 1 ∪ · · · ∪ η m ). We are then left with trees where each y has order where, as before, T m denotes the trees on {1, . . . , m} and (4.83) The latter sums over forests can be estimated replacing again L i by (4.62): Together with the link (x 1 , x 2 ) we obtain a tree on {1, . . . , m}: The sum over partitions can in fact be evaluated and yields Summing over n and k we have Note that this agrees with (4.69) in case m = 3.
For c ≥ 3 there is at least one point y ∈ γ which is connected to more than two other vertices. The tree on γ connecting the different components can again be reduced to a tree where all vertices have order ≥ 3 by integrating out the vertices y of order 2, yielding factors K (k) . The number of such vertices in γ is at most c − 2, where c is the number of components of η 1 ∪ · · · ∪ η m . The reduction formula reads (4.88) Here T (y 1 , . . . , y r ) is the set of tree graphs on r points, M (3) (T, c, r) denotes the set of maps π : {1, . . . , c} → {1, . . . , r} such that each point y i has at least 3 lines attached in the resulting graph, i.e. |{y ∈ T : (y i , y) ∈ T }| + |π −1 (i)| ≥ 3 for i = 1, . . . , r. π ∈ M (3) (T, c, r) determines the points of attachment of each component to the tree T . The sum over {I j } c j=1 is over partitions of {1, . . . , m} into c non-empty subsets, where 1 ∈ I 1 . I j is the set of i such that η i belongs to the j-th component.
The factor 1/k! compensates the factor k! in (4.78), and the factors k r and (k − r)! then count the number of ways of choosing which y i are associated with the vertices of T and the number of ways of distributing the remaining y i over the vertices of order 2.
If q is the number of vertices y ∈ γ of T connected to at least 3 other points of γ, then the tree T determines q − 1 paths between these vertices. In addition, there are q e ≥ 3q − 2(q − 1) = q + 2 endpoints. Each intermediate point of the tree must be connected to at least one components of η 1 ∪· · ·∪η m , whereas each endpoint must be connected to at least two. Let t be the number of intermediate points. Then c ≥ t + 2q e . It follows that r = q e + q + t ≤ 2q e − 2 + t ≤ c − 2.
The contribution to Q(η 1 ; . . . ; η m | k) of a given tree T ∈ T (3) with r vertices and an assignment π can be written as follows.
Q {Ij } c j=1 ,T,π (k) = h r r! dy 1 . . . dy r (k y,y ) (y,y )∈T : k y,y ≥0 (y,y )∈T k y,y ≤k−r (y,y )∈T : k j ≥0 c j=1 kj + (y,y )∈T k y,y =k−r so that This is bounded by (y,y )∈T k y,y =k−r (k j ) c j=1 : k j ≥0 c j=1 kj + (y,y )∈T k y,y =k−r × dy 1 . . . dy r (y,y )∈T ν(y − y ) As in the cases c = 1, 2 (see (4.84)), the factors fj ∈S(ηi j ;η I j \{i j } |∅) (x,x )∈fj ν(x−x ) can be bounded as follows Inserting this, we have (4.93) We have to bound the integral dy 1 · · · dy r (y,y )∈T where x j ∈ η Ij = ∪ i∈Ij η i and (y, y ) is a line in T between y i and y j for some i, j = 1, . . . , r. (N.B. It is allowed for y π(j) to be equal to y π(j ) with j = j . However, the number of unequal y π(j) must be at least twice the number of endpoints of the graph T on γ.) To estimate this integral, we integrate subsequently over the endpoints of T except the endpoint π(1) connected to I 1 .
Integrating over an endpoint y we have by Lemma 4.9: Therefore, setting p = |π −1 (i)|, Inserting this into (4.91), the first term in brackets combines trees T j on clusters connected to the same endpoint y i (i.e. π(j) = i) into a single tree T i connected to y i . The second term connects all x j with π(j) = i to y i . Note also that the factors C |Ij |−1 combine to give c j=1 C |Ij |−1 = C m−c and similarly, c j=1 (1 + C) i∈I j ; i =1 (li−1) = C l−l1−m+1 . Thus, where we write We can rewrite Q as follows, singling out the endpoints ∂T of T other than π(1), and denoting Λ(π, T ) = π −1 (∂T \ π(1)), Q {Ij } c j=1 ,T,π (r) = dy 1 . . . dy r (y,y )∈(T \∂T )∪π (1) ν(y − y ) ν(x j − y π(j) )ν(y π(j) − y π(j) ) max Tj ∈T (Ij ) ν Tj . After this first integration, some of the neighbours y i have become endpoints of a reduced tree T . We integrate out these points next and proceed this way until T is reduced to a single point. We can write the expression (4.99) in terms of the reduced tree as follows.
where for each i ∈ (T \ ∂T ) ∪ {π(1)}, the set of j such that π (j ) = i is given by where either S i = {j} for some j ∈ π −1 (i) or S i = π −1 (i). These two cases correspond respectively to the two terms in the last-but-one factor above (in (4.99)). In the first case, the trees T j with π(j) = i combine into a single tree In the second case, the forests f j are unchanged. The number of components is reduced to The corresponding subdivision is, in the first case I j = j ∈π −1 (i) I j , and in the second case I j = I j for all j ∈ π −1 (i). (Obviously, I j = I j for j ∈ π −1 (i ) in any case.) Moreover, we have modified the definition of Q {Ij } c j=1 ,T,π , replacing ν Tj byν T j = j ∈π −1 (i); j =j l I j ν T j if S i = {j}.
After at most r − 1 stages, the graph reduces to a single point r = 1. At the final stage we have to integrate over the last vertex y = y π(1) :   In this case, each term of the expansion is the sum of the contributions of the connected Cayley tree-graphs, and the expansion itself coincides with that obtained by Penrose in [29].