Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the dimension and $\alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)


Introduction 1.Motivation and overview
Since its invention in 1985 [16], the lace expansion has become a powerful tool for proving mean-field behavior in various spatial stochastic systems, such as the self-avoiding walk, percolation, oriented percolation, the contact process, lattice trees and -animals, and the Ising model.This paper provides a generalized lace expansion approach that holds for self-avoiding walk, percolation and the Ising model.We consider the classical nearest-neighbor model as well as various spread-out cases.Of particular interest are those spread-out models where the underlying step distribution has infinite variance, socalled long-range models.We show that a sufficiently long range can reduce the upper critical dimension, above which the system shows mean-field behavior.
We shall not perform the complete lace expansion here, but rather use bounds on the lace expansion coefficients proved elsewhere.Nevertheless, we give an analysis of the lace expansion inspired by [14], which is simplified compared to previous work, and generalized so that it deals with long-range models.
Using this generalized framework, we do the analysis of the lace expansion in such a way that it holds for any model provided that the expansion has a specific form and certain bounds on the lace expansion coefficients are satisfied (see Section 2).These bounds are proved to follow from a related random walk condition, which is relatively simple to verify.

The model
We study self-avoiding walk, percolation and the Ising model on the hypercubic lattice Z d .We consider Z d as a complete graph, i.e., the graph with vertex set Z d and corresponding edge set Z d × Z d .We will refer to the edges as bonds and to the vertices as sites.We assign each (undirected) bond {x, y} a weight D(x − y), where D is a probability distribution specified in Section 1.2.1 below.If D(x − y) = 0, then we can omit the bond {x, y}.
Our analysis is based on Fourier analysis.Unless specified otherwise, k will always denote an arbitrary element from the Fourier dual of the discrete lattice, which is the torus [−π, π) d .The Fourier transform of a summable function f : Z d → C is defined by f (k) = x∈Z d f (x) e ik•x .

The step distribution D: 3 versions
Let D denote a probability distribution on Z d that is symmetric under reflections in coordinate hyperplanes and rotations by π/2.We refer to D as a step distribution, having in mind a random walker taking independent steps distributed according to D. Without loss of generality we henceforth assume that there is no mass at the origin, i.e.D(0) = 0.
In this paper, we consider three different versions of D. While we explicitly state our main results for these versions, they actually hold more generally under a random walk condition formulated in Assumption 2.1 below.The first version is the nearest-neighbor model, where D is the uniform distribution on the nearest neighbors, i.e., x ∈ Z d . (1.1) Here, and throughout the paper, we denote by | • | the Euclidian norm on Z d and ½ E represents the indicator function of the event E. This nearest-neighbor version of D corresponds to the classical model for the study of self-avoiding walk, percolation, and the Ising model, see e.g.[21,24,34].We further consider two versions of spread-out models.They involve some spread-out parameter L, which is typically chosen large.In order to stress the L-dependence of D we will write D L in the definitions, but later omit the subscript.In the finite-variance spread-out model we require D L to satisfy the following conditions1 : (D1) There is an ε > 0 such that (D2) There is a constant C such that, for all L ≥ 1, (D3) There exist constants c 1 , c 2 > 0 such that Example.Let h be a non-negative bounded function on R d which is almost everywhere continuous, and symmetric under the lattice symmetries of reflection in coordinate hyperplanes and rotations by ninety degrees.Assume that there is an integrable function H on R d with H(te) non-increasing in t ≥ 0 for every unit vector e ∈ R d , such that h(x) ≤ H(x) for all x ∈ R d .Assume further that the (2 + ε)-th moment of h exists for some ε > 0. The monotonicity and integrability hypotheses on H imply that x h(x/L) < ∞ for all L, with x/L = (x 1 /L, . . ., x d /L).Then obeys the conditions (D1)-(D3), whenever L is large enough (cf.[32, Appendix A]).For h(x) = ½ {0< x ∞ ≤1} we obtain the uniform spread-out model with (1.6) In the spread-out power-law model we replace assumptions (D1) and (D3) by the condition that there exists an α > 0 such that (D1 ′ ) all ε > 0 satisfy (1.9) The condition (D2)=(D2 ′ ) remains unchanged.
As an example, let D L be of the form (1.5), but instead of the existence of the (2 + ε)-th moment of h, require h to decay as |x| −d−α as |x| → ∞.In particular, there exist positive constants c h and l h such that h(x) ≥ c h |x| −d−α , whenever |x| ≥ l h .(1.10)In this setting, the κ th moment x∈Z d |x| κ D L (x) does not exist if κ ≥ α, but exists and equals O(L α ) if κ < α.Take e.g.
so that D L has the form (1.12) Chen and Sakai [18,Prop. 1.1] showed that, analogously to the finite-variance spread-out model, the spread-out power-law model (1.12) satisfies conditions (D1 ′ )-(D3 ′ ).Note that the spread-out power-law model with parameter α > 2 satisfies the finite variance condition (D1), and hence is covered in the finite variance case.For simplicity we further write α ∧ 2 indicating the minimum of α and 2 in the spread-out power-law case, and 2 in the nearest-neighbor case or in the finite-variance spread-out case.
For the finite-variance spread-out model and the spread-out power-law model we require that the support of D contains the nearest neighbors of 0, see the discussion below (1.22).
We next introduce the models that we shall consider, i.e., self-avoiding walk, percolation and the Ising model.

Self-avoiding walk
For every lattice site x ∈ Z d , we denote by W n (x) = {(w 0 , . . ., w n ) | w 0 = 0, w n = x, w i ∈ Z d , 1 ≤ i ≤ n − 1} (1.13) the set of n-step walks from the origin 0 to x.We call such a walk w ∈ W n (x) self-avoiding if w i = w j for i = j with i, j ∈ {0, . . ., n}.We define c 0 (x) = δ 0,x and, for n ≥ 1, (1.14 where D is as in Section 1.2.1.

Percolation
In percolation we consider the set of bonds, which are unordered pairs of lattice sites.We set each bond {x, y} ∈ Z d × Z d occupied, independently of all other bonds, with probability zD(y − x) and vacant otherwise.Thus for the nearest-neighbor model, each nearest-neighbor bond is occupied with probability z/(2d).The corresponding product measure is denoted by P z with corresponding expectation E z .We require z ∈ [0, D −1 ∞ ] to ensure that zD(x − y) ≤ 1.We write {x ↔ y} for the event that there exists a path of occupied bonds from x to y.When the event {x ↔ y} occurs we call the vertices x and y connected.
It is the size and geometry of these clusters that we are interested in.Due to the shift invariance of the model, we can restrict attention to the cluster at the origin C := C(0).
For z small, C is P z -a.s.finite, whereas for d ≥ 2 and large z, the probability that the size of the cluster C is infinite, θ(z) is strictly greater than zero.Since z → θ(z) is non-decreasing, there exists some critical value z c where this probability turns positive (see e.g.[24]).

Ising model
For the Ising model we consider the space {−1, 1} Z d of spin configurations on the hypercubic lattice, with a probability distribution thereon.For a formal definition, we consider a finite subset Λ ⊂ Z d , and for every spin configuration ϕ = {ϕ x |x ∈ Λ} ∈ {−1, 1} Λ the energy given by the Hamiltonian where J and D are related via the identity and z is the inverse temperature.For example, in the nearest-neighbor case, D = J.For the Ising model, J is known as the spin-spin coupling.If J ≥ 0 (and hence D ≥ 0, as in the cases we consider) then the model is called ferromagnetic.

Two-point function and susceptibility
We study self-avoiding walk, percolation and the Ising model in a unified way.For this, we need to introduce some notation.We consider the function G z (x) for x ∈ Z d with being the Green's function for self-avoiding walk, while for percolation being the probability of the event that there is a path consisting of occupied edges from 0 to x.For the Ising model, we consider the spin correlation G z as the thermodynamic limit . (1.20) Here the limit is taken over any non-decreasing sequence of Λ's converging to Z d .This limit exists and is independent from the chosen sequence of Λ's due to Griffiths' second inequality [23].We will refer to G z as the two-point function.This is inspired by the fact that G z (x) describes features of the models depending on the two points 0 and x.
We further introduce the susceptibility as For percolation, the susceptibility is the expected cluster size χ(z) = E z |C|.
We define z c , the critical value of z, as For self-avoiding walk, z c is the convergence radius of the power series (1.18).For percolation, z c is characterized by the explosion of the expected cluster size.Menshikov [35], as well as Aizenman and Barsky [2], showed that this characterization coincides with the critical value described in Section 1.2.3.
For the spread-out models, we require that the support of D contains the nearest neighbors of 0. In percolation and the Ising model, this enables a Peierls type argument showing that that a (finite) critical threshold z c ∈ (0, ∞) exists, where the susceptibility χ(z) diverges as z ր z c .This is exemplified in [21, Sect.2.1] for the Ising model, and [24,Sect. 1.4] for percolation.
For the Ising model, we define the magnetization M to be and write M (z, 0 + ) for the limit lim hց0 M (z, h).The magnetization gives rise to another characterization of z c , namely z c = inf{z | M (z, 0 + ) > 0}.As proved by Aizenman, Barsky and Fernández [3], this is equivalent to (1.22).

Critical exponents and mean-field behavior
All three models, self-avoiding walk, percolation and the Ising model, exhibit a phase transition at some (model-dependent) critical value z c .One of the fundamental question in statistical mechanics is how models behave at and nearby this critical value.We use the notion of critical exponents to describe this behavior.While the existence of these critical exponents is folklore, there is no general argument proving this.We write f (z) ≍ g(z) if the ratio f (z)/g(z) is bounded away from 0 and infinity, for some appropriate limit.For self-avoiding walk, we define the critical exponents γ S and η S by For percolation we define the critical exponents γ P , β P , δ P and η P by The exponent γ P describes the asymptotic behavior in the subcritical regime {z < z c }, β P describes the behavior in the supercritical regime {z > z c }, and δ P and η P describe the behavior at criticality.For the Ising model, we consider the critical exponents γ I , β I , δ I , η I defined by For a discussion on the construction of Ĝzc (k) we refer to Section 2.1 below.It is believed that critical exponents are universal, i.e., minor modifications of the model, like changes in the underlying graph, leave the general asymptotic behavior, as described by the critical exponents, unchanged.Their values depend on the dimension d.However, it is predicted that there is an upper critical dimension d c , such that the critical exponents take the same value for all d > d c .These values are the mean-field values of the critical exponents.For self-avoiding walk these are the values obtained for simple random walk, i.e., γ S = 1 and η S = 0, whereas for percolation the mean-field values are γ P = 1, β P = 1, δ P = 2 and η P = 0, which coincide with the corresponding critical exponents obtained for percolation on an infinite regular tree, see [24,Section 10.1].For the Ising model, these mean-field values are γ I = 1, β I = 1/2, δ I = 3 and η I = 0, as obtained for the Curie-Weiss model.
The present paper uses the lace expansion to show that these critical exponents exist and take their mean-field values in sufficiently high dimensions for the nearest-neighbor version of D, or d exceeding some critical dimension d c and L sufficiently large for the spread-out models, respectively.

Results
We introduce the (small) quantity β by β = K/d for the nearest-neighbor model (K is a uniform constant), or β = K L −d for the spread-out models (K is a constant depending on d and α).We make this relation more explicit in Proposition 2.2 below.Be aware that the critical exponents β P and β I have no relation with the β introduced here.
We further introduce the function τ : z → τ (z), where τ (z) = z for self-avoiding walk and percolation, and τ (z) = (1.35) The infrared bound is well-known in several cases.Hara and Slade proved the infrared bound for the nearest-neighbor case and the finite-variance spread-out case, for self-avoiding walk [27,28] (see also [34,Theorem 6.1.6])as well as for percolation [26].Fröhlich, Simon and Spencer [22] proved the upper bound in (1.35) for the Ising model under the reflection positivity assumption, which holds e.g. for the nearest-neighbor case.We discuss reflection positivity in more detail in Section 1. 4.
By discarding the term χ(z) −1 in (1.35), we obtain from Theorem 1.1 that (under the assumptions formulated there) (1.36) .37)holds in all our three models: for self-avoiding walk this is obvious, for percolation it follows from the BK-inequality [11], and for the Ising model we use [38, (4.2)] in the infinite-volume limit.Thus for s = 2, A combination of (1.36) and (1.38) gives rise to where we use that the integrated term is O(β) by Assumption 2.1 and Proposition 2.2 below.A similar calculation gives the corresponding result for s = 3.More specifically, The two-point function G z (x) seen as a function of z (for fixed x) is continuous.For self-avoiding walk this fact follows from Abel's Theorem, and for percolation it is a consequence of Aizenman, Kesten and Newman [7].A general argument that holds for all our three models is the following: the quantity G z (x) can be realized as an increasing limit (finite volume approximation) of a function which is continuous and non-decreasing in z, hence G z (x) is left-continuous (cf.[25,Appendix A]).It follows that (1.38)-(1.40)even hold at criticality, i.e. when z = z c .In particular, this implies the bubble condition (i.e., B(z c ) < ∞) or the triangle condition (i.e., T (z c ) < ∞) for s = 2 or 3, respectively.We formulate this fact as a corollary: The bubble/triangle condition is important since it implies mean-field behavior of the model, which is formulated in the next theorem.In fact, (1.35) extends to the critical case z = z c as Ĝzc and we refer to the discussion around (2.7) below for a construction of Ĝzc (k) and a derivation of (1.41).We now use Theorem 1.1 to establish the existence of the formerly introduced critical exponents.(iv) For all three models, under the assumptions in Theorem 1.1 and if i.e., the critical exponents η S = η P = η I = 0 exist.
The derivation of the critical exponents from the bubble-/triangle condition (Corollary 1.2) is wellknown in the literature.However, the mode of convergence required for the existence of the critical exponents varies, and some derivations are stated only for finite range models.We therefore add a more detailed discussion of the literature here.
For self-avoiding walk, the existence (and the value) of the critical exponent γ S is based on the inequality The derivation of the exponents γ P = 1, β P = 1 and δ P = 2 from the triangle condition is due to Aizenman-Newman [8] and Barsky-Aizenman [10].To apply these results in our settings, there are some subtle issues to be resolved, and we discuss these in more detail in Appendix A.
For the Ising model, it has been proven by Aizenman [1, Proposition 7.1] that the bubble condition implies γ I = 1 as long as |J| = x J(x) < ∞ (which is equivalent to x D(x) < ∞).Under the same condition, Aizenman and Fernández [5] proved the existence and mean-field values of the critical exponents β I and δ I .
The statement in (iv) is an immediate consequence of (1.41).The lower bound in 1 − D(k) ≍ |k| α∧2 follows from (D3)/(D3').The upper bound indeed holds for a number of examples, and in particular if D is chosen as in the nearest-neighbor model (1.1), the finite-variance spread-out model (1.6) or the spread-out power-law model (1.12) with α = 2, cf.[18,32].However, if D is chosen as in (1.12) with The proof of Theorem 1.1, as well as the proof of Corollary 1.2, is given at the end of Section 2.

Discussion and related work
There is numerous work on the application of the lace expansion, see the lecture notes by Slade [41] and references therein.We give more references below at places where we use lace expansion methodology and need particular results.We now briefly summarize the results known for long-range systems.
Long-range self-avoiding walk has rarely been studied.Klein and Yang [42] showed that weakly self-avoiding walk in dimension d ≥ 3 jumping m lattice sites along the coordinate axes with probability proportional to 1/m 2 converges to a Cauchy process (as for ordinary random walk with such step distribution).A similar result for strictly self-avoiding walk has been obtained by Cheng [19].
For percolation, Hara and Slade [26] proved the infrared bound for the finite-variance spread-out case when D has exponential tails.The study of long-range percolation with power law spread-out bonds started in the 1980's by considering the one-dimensional case [9,36,39].These authors study the case where occupation probabilities are given by (1.12) with α ∈ (0, 1] and prove criteria for the existence of an infinite cluster.For example, Aizenman-Newman [9] show that if D(x) |x| 2 → 1 as |x| → ∞ in one dimension, and D(1) is sufficiently large, then there exists a critical infinite cluster and hence the percolation probability z → θ(z) is discontinuous at z c .This is compatible with our results, which imply that there is no infinite cluster at criticality for d > 3α (and here α = 1).Berger [12] uses a renormalization argument to show that in dimension d = 1, 2 the infinite cluster (if it exists) is transient if 0 < α < d and recurrent if α ≥ d.He further concludes that in the d-dimensional case (d ≥ 1) there is no infinite cluster at criticality if 0 < α < d.The question whether there exists an infinite critical cluster for d ≥ 2 and α ≥ d [12, Question 6.4] is answered negatively by the present paper for d > 6 and L sufficiently large.
In a recent paper, Chen and Sakai [18] study oriented percolation in the spread-out power-law case.Using similar methods, they prove that the two-point function in oriented percolation obeys an infrared bound if d > 2(α ∧ 2), which implies mean-field behavior of the model.
A long-range Ising model in one dimension has been studied by Aizenman, Chayes, Chayes, and Newman [4].Similar to the percolation result in [9], they prove that in the one-dimensional case where D(x) |x| 2 → 1 as |x| → ∞, the spontaneous magnetization M (z, 0+) has a discontinuity at the critical point z c .
The infrared bound for the Ising model was proved in [22] for d > α ∧ 2 for a class of models obeying the reflection positivity (RP) property.The class of models satisfying (RP) includes the nearest-neighbor model (where D(x) = (2d) −1 ½ {|x|=1} ), exponential decaying potentials (where D(x) ∝ exp{−µ x 1 } for µ > 0), power-law decaying interactions (where D(x) ∝ |x| −s for s > 0), and combinations thereof.For a definition of (RP) and a discussion of the above mentioned models, we refer to [13].Nevertheless, (RP) fails in most cases for small perturbations of these models, although it is believed that the asymptotics still hold.Moreover, (RP) only implies the upper bound in (1.35), in that implying that the critical exponent η (when it exists) is nonnegative.Our approach using the lace expansion does not require reflection positivity, it is much more universal in the choice of D (cf.Section 1.2.1), and also gives a matching lower bound in (1.35), yielding η = 0. On the other hand, our approach requires that the dimension d or the spread-out parameter L are sufficiently large, a limitation that one may not expect to reflect the physics.The literature for the long-range Ising model in higher dimensions based on (RP) arguments is summarized by Aizenman and Fernández [6], who also identify 2(α ∧ 2) as upper critical dimension. 2iven (1.42) it is folklore that holds in the general setting considered here.Partial results towards (1.44) have been obtained.Indeed, Hara, van der Hofstad and Slade [29] proved (1.44) in the finite-range spread-out setting for self-avoiding walk and percolation, Hara [25] proved it in the nearest-neighbor setting, and Sakai [38] proved it for the Ising model in finite-range spread-out and nearest-neighbor settings.We discuss the critical two-point function G zc (x) at the end of Sect.2.1.

A general framework
In order to study the various models in a unified way, we use this section to set up a generalized framework.We make two assumptions in terms of the general framework, and use the subsequent two sections to show that our models actually satisfy these assumptions.We then prove the results within the abstract setting, based on the two assumptions made.

An expansion of the two-point function
Given a step distribution D, we consider the random walk two-point function or Green's function of the random walk defined by where D * n is the n-fold convolution of D and D * 0 (x) z 0 = δ x,0 .We write δ for the Kronecker delta function.By conditioning on the first step we obtain Taking the Fourier transform and solving for Ĉz (k) yields Next we consider G z (x) defined in (1.18)- (1.20).For each of the three models, i.e., for self-avoiding walk, percolation and the Ising model, we use the lace expansion to obtain an expansion formula of the form The coefficients Φ z (x) and Ψ z (x) depend on the model, but above their respective upper critical dimension they obey similar bounds.Assuming the existence of Φz (k) and Ψz (k), Fourier transformation yields Ĝz The full derivation of the lace expansion will not be carried out in this paper.We discuss the lace expansion briefly in Section 4, where we also define the lace expansion coefficients Φ z and Ψ z , and cite bounds on them from [14,38,41].We will see that, for z = 0, Ψ0 (k) ≡ 0 and Φ0 (k) ≡ 0 for all models.We recall that τ (z) = z for self-avoiding walk and percolation, and τ (z) = y∈Z d tanh(zJ(y)) for the Ising model, see Sect.1.3.For the critical case (i.e., z = z c ) we have where the lower bound is a consequence of (1.37), and the upper bound emerges from (2.18) and (2.29) below.The function An issue of interest is the (left-) continuity of Ĝz (k) at z = z c .In particular, the identity ), but a proof for the Ising model is not known.

The random walk condition
Recall that the model parameter s is 2 for self-avoiding walk or Ising model, and 3 for percolation.We now make an assumption on the step distribution D.

Remark:
The specific amount of smallness required in (2.9)-(2.10)will be specified in the proofs in Section 5.
For s = 2 we call (2.10) the random walk bubble condition.This is inspired by the fact that its x-space analogue reads In other words, we have an (ordinary) random walk from 0 to x of at least one step, and a second walk from x to 0 and subsequently sum over all x.Correspondingly, for s = 3, we obtain the x-space and refer to (2.10) as the random walk triangle condition.See the graphical representation in Figure 1.
We remark that Theorem 1.3 generalizes in the same way.

Diagrammatic bounds
We introduce the quantity Then λ z satisfies the equality Ĝz (0) = Ĉλz (0). (2.14) The idea of the proof of Theorem 2.3 is motivated by the intuition that Ĝz (k) and Ĉλz (k) are comparable in size and, moreover, the discretized second derivative is bounded by More precisely, we will show that the function f : [0, z c ) → R, defined by and is small, given that β in Assumption 2.1 is sufficiently small.To make this rigorous, we need the following assumption: Assumption 2.4 (Bounds on the lace expansion coefficients).If, for some K > 0, the inequality f (z) ≤ K holds uniformly for z ∈ (0, z c ), then there exists a constant c K > 0 such that, for all where Φ z and Ψ z refer to the model-dependent coefficients in the expansion formula (2.4).
The key to our results is that the bounds (2.20)-(2.21)imply Theorem 2.3 (and hence Theorem 1.1):
The relevant bounds have been proven by Slade [41] for self-avoiding walk, by Borgs et al. [14] for percolation (on finite graphs), and by Sakai [38] for the Ising model.In Section 4 we state the diagrammatic bounds proved in these papers, and relate them to our version of Φ z and Ψ z , thus proving Proposition 2.5 using [14,38,41].

Completion of the argument and organization of proofs
The proof of Theorem 2.3 will follow from the following proposition: Proposition 2.6.Suppose we are given a model with some model-dependent constant s ∈ {2, 3, . . .}, and a two-point function G z of the form (2.4), where the step distribution D satisfies Assumption 2.1, and Φ z and Ψ z satisfy Assumption 2.4, both for the same sufficiently small β > 0. Assume further that uniformly for z < z c .
The assumption χ ′ (z) ≤ const χ(z) 2 in Proposition 2.6 can be replaced by assuming that f is continuous on [0, z c ), cf.Lemma 5.3 below.It is known as a mean-field bound, and a proof of it can be found in [41,Theorem 2.3] for self-avoiding walk, and in [41, Prop.9.2] for percolation.For the Ising model, this mean-field bound is a consequence of the Lebowitz inequality [33].
In order for Theorem 2. The remainder of the paper is organized as follows.In Section 3 we prove Proposition 2.2 by showing that Assumption 2.1 is satisfied for our versions of D. For the proof of Proposition 2.5 we need the lace expansion.The diagrammatic bounds are not derived in the present paper; instead we explain in Section 4 how to obtain the statement of Proposition 2.5 from the diagrammatic bounds in [41] for self-avoiding walk, [14] for percolation, and [38] for the Ising model.Finally, the proof of Proposition 2.6 is contained in the last Section 5, and this completes the proof of Theorem 2.3 (and hence of Theorem 1.1 and Corollary 1.2).Appendix A contains a derivation of the existence and the mean-field values of the critical exponents γ P and δ P for percolation.In Appendix B we show how the bounds on the lace expansion in Assumption 2.4 for the Ising model can be obtained from the diagrammatic bounds in [38].Our account in Appendix B follows the proof of [38,Prop. 3.2], but with a modified bootstrap hypothesis.

The random walk two-point function
In this section we prove Proposition 2.2.The estimates below are contained in [14, Sect.2.2.2],where finite tori are considered.Restriction to the infinite lattice gives rise to a noteworthy simplification, which we shall present in the following.
By the symmetry of D we have The Cauchy-Schwarz inequality3 yields First we show that the first term on the right hand side of (3.3) is small if d is large.Note that is the probability that a nearest-neighbor random walk returns to its starting point after the fourth step.This is bounded from above by c(2d) −2 with c being a well-chosen constant, because the first two steps must be compensated by the last two.Finally, the square root yields the upper bound O(d −1 ).
It remains to show that the second term on the right of (3.3) is bounded uniformly in d.The infrared bound (3.2) gives The right hand side of (3.4) is finite if d > 4s.For A > 0 and m > 0, as an upper bound for (3.4).This is non-increasing in d, because f p ≤ f q for 0 < p ≤ q ≤ ∞ on a probability space by Lyapunov's inequality.
The power-law spread-out model with α > 2 satisfies the finite variance condition (D1) with ε < α − 2. Note further that (D3) and (D3 ′ ) agree when the exponent in the first inequality is taken α ∧ 2.
We separately consider the regions k ∞ ≤ L −1 and k ∞ > L −1 .By (1.2), (1.7) and the bound D(k) 2 ≤ 1, the corresponding contributions to the integral are for some positive constant.In the last step we used assumption (D2) / (D2 ′ ) to see that

The lace expansion
In this section, we discuss the lace expansion which obtains an expansion of the two-point function of the form The key point is to identify the lace-expansion coefficients Φ z and Ψ z in a way that allows for sufficient bounds, known as diagrammatic bounds.The derivation is not carried out in this paper; full expansions and detailed derivations of the diagrammatic bounds are performed in [31,41] for selfavoiding walk, in [14] for percolation and in [38] for the Ising model.

The lace expansion for the self-avoiding walk
The lace expansion for self-avoiding walks was first presented by Brydges and Spencer [16].They provide an algebraic expansion using graphs.A special class of graphs that play an important role here, the laces, gave the lace expansion its name.An alternative approach is based on an inclusion-exclusion argument, and was first presented by Slade [40].
We refer the reader to [31, Sect.2.2.1] or [41,Sect. 3] for a full derivation of the expansion.For example, in [31, Sect.2.2.1] it is shown that for suitable π m (x).We multiply (4.1) by z n+1 and sum over n ≥ 0. By letting see also [41, (3.27)].For the lace expansion coefficient Π z the following diagrammatic bound is proven: Proposition 4.1 (Diagrammatic estimates for self-avoiding walk from [41]).Fix z ∈ (0, z c ).If f (z) of (2.17) obeys f (z) ≤ K, then there are positive constants c K and β 0 = β 0 (K), such that the following holds: If Assumption 2.1 holds for some β ≤ β 0 , then The term diagrammatic estimate originates from the fact that Π z is expressed in terms of diagrams.The underlying structure expressed in terms of these diagrams is heavily used to obtain the bounds in (4.3) and (4.4).
A proof of Prop.4.1 can be found in [41,Lemma 5.11], and we do not repeat it here.In fact, the proof in [41] can be modified to obtain instead of (4.4).This is achieved by leaving the factor z in [41, (5.42) and (5.43)] explicit (rather then bounding above by K).

The lace expansion for percolation
The lace expansion for percolation was first derived in [26].It is based on an inclusion-exclusion argument, and holds quite generally for any connected graph, finite or infinite.The graph does not even need to be transitive or regular.
In [14,Sect. 3.2], the identity is derived for M = 0, 1, 2, . . . .The z-dependence of Π M and R M is left implicit.The function Π M : Z d → R is the central quantity in the expansion, and R M (x) is a remainder term.When the expansion converges, one has lim The subscript M denotes the level to which the (inclusion-exclusion) expansion is carried out, and we shall later fix M so large that (4.12) and (4.13) below are satisfied for K = 4.The equality (4.6) is equivalent to (2.4) if we let τ (z) = z, and The key point is that Π M and R M satisfy useful diagrammatic bounds: Proposition 4.2 (Diagrammatic estimates for percolation from [14]).Fix z ∈ (0, z c ).If f (z) of (2.17) obeys f (z) ≤ K, then there are positive constants c K and β 0 = β 0 (K), such that the following holds: If Assumption 2.1 holds for some β ≤ β 0 , then for all M = 0, 1, 2, . . ., x and for M sufficiently large (depending on K and z), x In fact, the bounds in Proposition 4.2 are not exactly as phrased in [14].In the following we explain how the proof of [14, Prop.5.2] can be modified to obtain Prop.4.2.There are two differences to consider.First, in the definition of f 3 there is a factor 16 in the denominator, whereas we have a factor 200, cf.(2.19).This can be controlled easily by changing the factor appropriately throughout the proof of [14,Prop. 5.2].This changes the specific value of c K , but the statement of [14, Prop.5.2] remains unchanged.The second (and more important) issue is the replacement of 1 − D(k) = Ĉ1 (k) −1 in [14, Prop.5.2] by 1 − λ z D(k) = Ĉλz (k) −1 in Prop.4.2.We need to do this replacement to achieve continuity of the function f 3 .Wherever the bound on f 3 is used in the proof of [14, Prop.5.2], which is in [14, (5.63)], [14, (5.77)], below [14, (5.93)] and in [14, (5.97)], we replace the factor [1 − D(k)] by Ĉλz (k) −1 .Other occurrences of [1 − D(k)], as in [14, (5.75)] and [14, (5.91)], can be treated with the bound which itself is a consequence of For the bounds on Φ z (x) = z(D * Π M )(x) we use the estimate (see [14, (4.51)]) to obtain

The lace expansion for the Ising model
The lace expansion for the Ising model has been established recently by Sakai [38].It is similar in spirit to a high-temperature expansion.A key point is to rewrite the two-point function (spin-spin correlation) using the random-current representation.This gives rise to a representation involving bonds, in that showing some similarities to a percolation configuration.The lace expansion is then performed using ideas from the lace expansion for percolation, however, it is considerably more involved.
For the Ising model on a finite graph Λ, Sakai in [38, Prop.1.1] proved the expansion formula where the z-dependence of Π Λ M and R Λ M is omitted from the notation.Note that R Λ M in this paper is (−1) M +1 R (M +1) p;Λ in [38].Here M refers to the level of the expansion, and G Λ z denotes the finite-volume two-point function.This is equivalent to (2.4) if we let then choose M so large that (4.23) and (4.24) below are satisfied for a certain K, say K = 4, and subsequently taking the thermodynamic limit Λ ր Z d .Note that, if comparing (4.18) to [38, (1.11)], we explicitly extract the δ 0,x -term from the Π-term in [38], i.e., Π (M ) p;Λ (x) in [38] corresponds to Π Λ M (x) + δ 0,x in this paper.For Π Λ M and R Λ M we have the following bounds: Proposition 4.3 (Diagrammatic estimates for the Ising model from [38]).Fix z ∈ (0, z c ).If f (z) of (2.17) obeys f (z) ≤ K, then there are positive constants c K and β 0 = β 0 (K), such that the following holds: If Assumption 2.1 holds for some β ≤ β 0 , then for all M = 0, 1, 2, . . ., x and for M sufficiently large (depending on K and z), x These bounds hold uniformly in Λ.
Since the bootstrapping hypothesis used in Section 5 in this paper is different from that in [38], it is not so obvious how Prop.4.3 follows from the results in [38].In Appendix B we explain how the statement in [38,Prop. 3.2] can be modified to obtain the desired bounds (4.21)-(4.24).
We prove Prop.2.5 for the Ising model as in the percolation case, now using Prop.4.3 instead of Prop.4.2.We refrain from repeating the argument.

Analysis of the lace expansion 5.1 The bootstrap argument
In this section we prove Proposition 2.6 and, by doing so, complete the proof of Theorem 1.1.The proof is based on the following lemma: Lemma 5.1 (The bootstrap / forbidden region argument).Let f be a continuous function on the interval [0, z c ), and assume that f (0 Proof.This is a straightforward application of the intermediate value theorem for continuous functions, see also [41,Lemma 5.9].
The bootstrap argument in Lemma 5.1 is often used in lace expansion, see e.g.[34,Section 6.1].An alternative approach that involves an induction argument has been applied in [32], see also the lecture notes by van der Hofstad [31].
In the remainder of the section, we prove that the function f defined in (2.17) obeys the prerequisites of Lemma 5.1.We therefore have to show that f (0) ≤ 3, that f is continuous on [0, z c ), and that f (z) ≤ 4 implies f (z) ≤ 3 for z ∈ (0, z c ).The latter is referred to as the improvement of the bounds.
Next we want to prove continuity of f .To this end, we need the following lemma: Lemma 5.2 (Continuity of equicontinuous functions).Let (f α ) α∈A be an equicontinuous family of functions on an interval [t 1 , t 2 ], i.e., for every given ε > 0, there is a A proof of this standard result can be found e.g. in [41,Lemma 5.12].
Proof.It is sufficient to show that f 1 , f 2 and f 3 are continuous.The continuity of f 1 is obvious.We show that f 2 and f 3 are continuous on the closed interval [0, z c − ε] for any ε > 0 by taking derivatives with respect to z and bound it uniformly in k on [0, We do f 2 first.To this end, we consider the derivative We proceed by showing that each of the terms on the right hand side is uniformly bounded in k and z ∈ [0, z c − ε], and hence the derivative is bounded.First we recall the definition of λ z in (2.13) to see that Furthermore, χ(z) ≤ χ(z c − ε), and the latter is finite by the definition of z c in (1.22).For every k ∈ [−π, π) d , the two-point function is bounded from above by For the derivative of the two-point function, we bound (5.4) where the exchange in the order of sum and derivative is validated by the fact that both x e ik•x G z (x) and x G z (x) are uniformly convergent series of functions.By the assumed mean-field bound χ ′ (z) ≤ cχ(z) 2 , (5.4) is bounded above by cχ(z c − ε) 2 .Moreover, we obtain from (2.3) that |d Ĉλ (k)/dλ| ≤ Ĉλ (k) 2 , and, for λ = λ z , this is in turn bounded by χ(z c − ε) 2 , cf. (5.2).Finally, |dλ z /dz| = χ ′ (z)/χ(z) 2 ≤ c by (2.13) and our assumption.
We treat f 3 in exactly the same way as f 2 , and omit the details here.

Improvement of the bounds
The following lemma covers the remaining prerequisite of Lemma 5.1 and thus proves the final ingredient needed for the proof of Proposition 2.6.
Lemma 5.4 (Improvement of the bounds).If the assumptions of Proposition 2.6 are satisfied for some sufficiently small β, and if f (z) ≤ 4, then there exists a constant c > 0 such that f (z) ≤ 1 + cβ for all z ∈ (0, z c ).In particular, if β is small enough, then f (z) ≤ 3.
The following lemma will help us for the improvement of the bound on f 3 .
Lemma 5.5 (Slade [41]).Suppose that a(x) = a(−x) for all x ∈ Z d , and let . (5.5) By |a| we denote the Fourier transform of the absolute value of a.The proof of Lemma 5.5 uses several bounds on trigonometric quantities, and can be found in [41,Lemma 5.7].
The bound on f 1 is easy.First note that λ z = 1 − χ(z) −1 ≤ 1.Using (2.13) along with (2.22)-(2.25)and Proposition 2.5 (with K = 4) we obtain (5.7) The bound on f 2 is slightly more involved.We write Ĝz = N / F , with and, by (2.5) and (2.13), This yields Ĝz (k) where By taking c 4 β ≤ 1/2, we obtain the bound which we use frequently below.For example, together with Assumption 2.4, it enables us to bound Together with (4.14) we obtain in the same fashion that By our assumption that Ĝz (k) ≤ 4 Ĉλz (k) (which follows from f (z) ≤ 4) and the above inequalities, we can bound (5.10) from above by Ĝz (k) (5.13) A straightforward calculation (see also [18, (4.18)]) shows that

.14)
We now bound all three summands in (5.14), and start with the first one: where the last bound uses (2.21) to bound the denominator, and (5.12).A basic calculation shows that any function g : cf. [14, (5.32)].We apply this bound with g(x) = Ψ z (x), combine it with (5.15) and (2.21), and use Ĉλz (l ± k) ≥ 1/2 and the definition of U λz (l, k) in (2.16) to obtain The second term in (5.14) is bounded as follows.First, since The second term on the right hand side of (5.19) is bounded by O(β) Ĉλz (k) −1 ; on the first term we apply the Cauchy-Schwarz inequality and (2.20)-(2.21): In a similar fashion as ( where the last line uses (4.14).The combination of (5.19)-(5.22)and (5.7) yields On the other hand, by (5.12)-(5.13), Combining (5.23) and (5.24) yields For the third term in (5. by Assumption 2.4 and (5.12), and where the last line uses again (4.14) and, as usual, requires a certain smallness of β (here we need c 4 β ≤ 1).Plugging these estimates into (5.6)yields ) so that finally as required.In conclusion f 3 (z) ≤ 1+Kβ, and thus we obtain the improved bound f (z) ≤ 1+O(β).
Proof of Proposition 2.6.Note first that f is continuous on (0, z c ) by Lemma 5.3 and the assumed mean-field bound χ(z) ′ ≤ const χ(z) 2 .Whence the prerequisites of Lemma 5.1 are satisfied by Lemma 5.4 and the fact that f (0) = 1.Therefore, f (z) ≤ 3 for all z < z c .Moreover, Lemma 5.4 shows that, if

A Derivation of critical exponents for percolation
A.1 Derivation of γ P = 1 Aizenman and Newman [8] prove that the triangle condition T (z c ) < ∞ implies that the critical exponent γ P for percolation exists, and satisfies γ P = 1.That is to say, they show χ(z) ≍ (z c − z) −1 as z ր z c .The lower bound γ P ≥ 1 in [8, Prop.3.1] holds for any homogeneous bond percolation model.On the other hand, the upper bound γ P ≤ 1 is stated in [8,Prop. 3.1] for the nearest neighbor model only.The aim of this section is to show how the derivation in [8] can be extended to long range systems.The argument requires a finite volume and range approximation in order to apply Russo's formula.We denote by a cube of sidelength 2r + 1.In order to achieve translation invariance, we equip the cube with periodic boundary conditions, that is, T r is a torus.In [8] free boundary conditions were used.We write G (R) z,Tr (x, y) for the probability that the points x and y are connected on the torus using only bonds {u, v} of length |u − v| ≤ R. For r > R (which we always assume), this is equivalent to removing all bonds from T r with length larger than R. Define accordingly the restricted expected cluster size by and the restricted triangle diagram by Integration over the interval (z, For the lower bound in (A.3) we use arguments as in [41,Section 9.4] to obtain (The contribution to the second line in (A.14) with u and v interchanged is hidden there, but is incorporated in the next line when we sum over both, u and v.) With Russo's formula (A.8), for z < z c − ε, and an integrated version of this proves (A.3).We now consider (A.4) and fix z < z c − ε.We write E (R) z,Tr |C| for the expected cluster size under the measure P (R)  z,Tr , i.e., E (R) z,Tr |C| = χ (R) Tr (z).We further denote by ∂ R T r := T r+R \ T r the boundary of T r of thickness R. Hence, In the first summand, E (R) z,T r+R can be replaced by E (R) z (the expected cluster size on the infinite lattice, where bonds are restricted to have length ≤ R), because the indicator guarantees C ⊂ T r .This leads to By the tree graph bound [8] and the monotonicity of and hence the Cauchy-Schwarz inequality yields For z < z c − ε, the first factor on the right is finite, and the latter vanishes as r → ∞.For the last summand in (A.18), we bound as follows: and hence The lower bound is more involved.For every ε > 0 we obtain 1 − e −εk/n P zc (|C| = k).
We exploit 1 − e −x ≤ x to bound further We apply (A.26) and compare with (A.25) to obtain This proves that , and c > 0 as long as ε is small enough.With a modification in (A.27), the argument can be extended to the case δ P = 2, but we refrain from giving this argument.

B Diagrammatic bounds for the Ising model
This appendix is devoted to the proof of Proposition 4.3 for the Ising model.We proceed by considering the quantities π (M ) Λ (M = 0, 1, 2, . . . ) defined in [38], which give rise to Π Λ M and R Λ M +1 by [38, (1.12) and (1.13)]: We first discuss a bound on π (N ) Λ , and use this to prove Proposition 4.3.Proposition B.1 (Diagrammatic bounds for the Ising model).Suppose that, for the Ising model, f (z) ≤ K for some z ∈ (0, z c ), K > 1.Then there exists a constant cK > 0, such that This proposition is a variation of [38,Proposition 3.2].However, it is important that the bounds of the type x |x| 2 π (N ) Λ (x) in [38] have been replaced by bounds involving the factor 1 − cos(k • x), as in (B.3).This replacement is a basic philosophy for this paper.The following heuristic reasoning explains why the factor |x| 2 is not sufficient in the case of infinite variance spread-out models.
For the first term, we use (4.14) and (B.2) to bound For the second term, we use (B.3) to see that π(M) . Finally, for the third term in (B.5), we use the upper bound on f 3 and the uniform bound Ĉλz (k) Together with (B.2), this yields the desired bound.
We now prove Proposition B.1 subject to the diagrammatic bounds in [38], which will occupy the remainder of the paper.Our proof is an adaptation of the proof of [38,Prop. 3.2], with a modified bootstrap hypothesis.In particular, the factor |x| 2 at various places in that proof is replaced by the factor 1 − cos(k • x) here.We fix z ∈ (0, z c ) and throughout the remainder of the section omit it from the notation (e.g., we write τ for τ (z)).Also we fix some subset Λ containing the origin.We keep in mind that we are interested in the thermodynamic limit Λ ր Z d , and in fact our bounds hold uniformly in Λ.We elaborate on this after Prop.B.2 below.All sums below are taken over Z d , unless stated otherwise. We Furthermore, it is easy to see that, by the Cauchy-Schwarz inequality, "open bubbles" are bounded by a "closed bubble", i.e., for all x ∈ Z d , Here is an outline of the proof.We bound certain diagrams to be defined below in terms of B and B. In turn, these diagrams bound the lace expansion coefficients π (j) , [38].Hence, by exploiting (B.9) and (B.9), we prove a sufficient decay of the lace expansion coefficients subject to β being sufficiently small.
We now define various quantities needed to describe the bounding diagrams.All notation is chosen consistently with [38], which provides our basic estimates.In order to emphasize the diagrammatic structure, we write G and G with two arguments, with the understanding that G(y, x) = G(x − y), and for G appropriately. Let denote a "chain of bubbles", and ψ(y, x) = ψ(y, x) − δ y,x .(B.12) If β is so small that B < 1/2 (which we shall assume from now on), then a basic calculation shows that ψ := sup  It should be noted that Sakai [38] proved the bound (B.26) on a finite graph Λ, where in particular all quantities on the right hand side are defined on Λ.By Griffith's second inequality [23], the two-point correlation function G z is monotonically increasing in Λ, and thus so are P ′ , Q ′ and Q ′′ .Hence, the right hand side in (B.26) is monotonically increasing in Λ, and we consider the thermodynamic limit Λ ր Z d as a uniform upper bound on π (N) Λ (x).However, it is not obvious how to obtain the thermodynamic limit on the left hand side directly, since the quantities π (N) Λ (x) are not monotone in Λ. Proof of (B.2).We first show that 1 . By the definition of π (0) Λ (x) and (B.14 The term x =0 G2 (x) is bounded above by a non-vanishing bubble B, yielding a factor O(β) by (B.9).The term sup x =0 G(x) can be bounded as follows.We first apply (1.37), to obtain The first summand is bounded by Kβ, by our bound on f 1 and (2.9).Furthermore, τ D * G ∞ ≤ 4K 3 β by a calculation similar to (B.9) and using 1 ≤ 2[1 − D(k)] −1 .We thus obtain the bound on x π (0) Λ (x).We next consider the bound on x π (N) Λ (x) for N ≥ 1.Here is a diagrammatic representation of the bounds on x π • • • (expression as above with x 3 fixed)   for j = 0, 1, 2, . . . .We explicitly perform this bound for j = 0, 1, and omit the details for j ≥ 2. This completes the proof of (B.2).
Proof of (B.3).We now turn towards the proof of the bound (B.3) in Proposition B.1, which we restate here for convenience: x We start by considering the case N = 0.By (B.26) and (B.14), x Then the desired bound follows from (B.9) and the following lemma: from [14, (4.51)], which is reminiscent of the decomposition of squares in [38, (5.39)].
In the case N = 1 this allows for the following calculation.Recall from Prop.B.2 the upper bound on π (1)  Λ (x).An application of (B.57) for N = 1 yields x [1 − cos(k • x)]π (1)  Λ (x) ≤ Here the dashed arrow indicates that the supremum is taken over the difference between the two vertices at top and bottom of the arrow; see also [38, (5.46)].In order to achieve the bounds in (B.74) we proceed as follows.

. 40 )
The bounds(1.38)-(1.40)hold uniformly for z < z c under the assumptions in Theorem 1.1.Note that in (1.40) we write G z (x, y) = G z (x−y).We call B(z) the bubble diagram and T (z) the triangle diagram.
) would follow from the the fact that Ψz (k) and Φz (k) are left-continuous at z = z c , as explained by Hara [25, Appendix A].The left-continuity of Ψz (k) and Φz (k) at z = z c indeed holds for self-avoiding walk (by Abel's Theorem) and for percolation (by [25, Lemma A.1]

35 )≤ B 4 39 )
For the remaining component on the right hand side, we again use translation invariance and bound further ψ4 , (B.37) and this can be made smaller than O(β) 4 , cf. (B.9) and (B.13).However, if the extra vertex falls to one of the vertical lines, then the details are slightly different: in (B.38) is bounded by multiple use of translation invariance, as we will This proves (B.33) for j = 4.The cases j ∈{0, 4} are omitted, since the same methods will lead to the desired bounds.(iii) We now turn to the bounds involving Q ′′ , i.e., we prove sup y w,v,x τ D(w − y)Q ′′ 0,v (w, x) ≤ O(β).(B.40) Recalling the definition of Q ′′ in (B.24), (B.40) is established once we have shown sup y w − y) δ w,z + G(w, z) P ′′ 0,v (z, x) ≤ O(β).(B.42)A decomposition of the left hand side of (B.41) yields as an upper bound  B.43) where the first term is bounded by O(β), the second term is bounded by 1 + ψ = O(1) and the final term is bounded by O(1), by Claim B.3.It thus remains to show the following claim: Claim B.4 (Bound on P ′′ ).The estimate (B.42) is true.Proof.In our pictorial representation, (B.42) can be expressed like ≤ O(β).(B.44) Similarly to the proof of (B.32), it is sufficient to show sup y w,v,z,x τ D(w − y) δ w,z + G(w, z) P ′′(j) 0,v (z, x) ≤ O(β) j∨1 (B.45)
hence the Fourier transform does not exist.However, diagrammatic bounds of the lace expansion coefficients (Prop.2.5) and the dominated convergence theorem guarantee the absolute convergence of the various sums involved defining Ψz (k) and Φz (k), which shows that the critical quantities Ψzc (k) and Φzc (k) are well-defined.This justifies the introduction of Ĝzc (k) as a solution to (2.5) with z = z c .Note that we do not assume any continuity of z → Ψz (k) and z → Φz (k) to do this.Nevertheless, we can extend (1.35) to the critical case z = z c , and further use (2.6) to obtain Ĝzc Figure 1: Graphical representation of the random walk bubble diagram in (2.11) and the random walk triangle diagram in (2.12).A line between two points, say x and y, represents the two-point function C 1 (y − x), a line with a double dash in the middle requires at least one step, e.g. a line between 0 and x represents (D * C 1 )(x).Vertices labeled in brackets are summed over Z d .Proposition 2.2.Assumption 2.1 is satisfied for arbitrarily small β if d is chosen sufficiently large in the nearest-neighbor model (at least d > 4s) or d > d We prove Proposition 2.2 in Section 3. We shall prove the following generalized version of Theorem 1.1.By Proposition 2.2, Theorem 2.3 below immediately implies Theorem 1.1.Theorem 2.3.Fix s = 2 for self-avoiding walk and the Ising model, and s = 3 for percolation.If Assumption 2.1 is satisfied for β sufficiently small, then (1.35) holds uniformly for c = s(α ∧ 2) and L is sufficiently large in the spread-out models.More specifically, the assumption holds with β = O(d −1 ) in the nearest-neighbor case, and β = O(L −d ) in the spread-out cases.
3 (and hence Theorem 1.1 and Corollary 1.2) to hold, we need to show (2.20)-(2.21).Indeed, (2.20)-(2.21)follow from the statements above, as we explain now.Propositions 2.2 and 2.5 validate Assumptions 2.1 and 2.4.With these assumptions, the prerequisites of Proposition 2.6 are satisfied and (2.29) holds for β sufficiently small by Proposition 2.2.The latter can be achieved by taking d or L large enough.Then we again use Assumption 2.4 to obtain (2.20)-(2.21),thus proving (1.35).
where C d,c 1 is a constant depending (only) on d and c 1 , and by (1.3), (1.8), In the following we show how Proposition 4.2 implies Proposition 2.5 in the percolation case.Proof of Proposition 2.5 for percolation.Recall (4.8)-(4.9).The bounds on Ψ z (x) in (2.20)-(2.21)follow directly from Proposition 4.2 if M is chosen so large that (4.12)-(4.13) is satisfied.
.15) Again, this increases the value of the constant c K , but leaves the statement of [14, Prop.5.2] otherwise unchanged.For a sketch of the argument of how f (z) ≤ K actually implies (4.10)-(4.13)we refer to [37, Sect.3.2].
By multiple use of translation invariance of the model, we obtain = x 1 ,x 2 ,x 3 ,x 4 , x 5 ,x 6 ,x 7 ,x 8