Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
aa r X i v : . [ m a t h . P R ] M a y Mean-field behavior for long- and finite rangeIsing model, percolation and self-avoiding walk
Markus Heydenreich , Remco van der Hofstad and Akira Sakai Eindhoven University of Technology,Department of Mathematics and Computer Science,P.O. Box 513, 5600 MB Eindhoven, The Netherlands [email protected], [email protected] Hokkaido University,Creative Research Initiative “Sousei”,North 21, West 10, Kita-kuSapporo 001-0021, Japan [email protected] (revised May 23, 2008)
Abstract:
We consider self-avoiding walk, percolation and the Ising model with long andfinite range. By means of the lace expansion we prove mean-field behavior for these modelsif d > α ∧
2) for self-avoiding walk and the Ising model, and d > α ∧
2) for percolation,where d denotes the dimension and α the power-law decay exponent of the coupling function.We provide a simplified analysis of the lace expansion based on the trigonometric approachin Borgs et al. [14]. MSC 2000.
Keywords and phrases.
Lace expansion, Ising model, percolation, self-avoiding walk, critical exponent,mean-field behavior.
Since its invention in 1985 [16], the lace expansion has become a powerful tool for proving mean-fieldbehavior in various spatial stochastic systems, such as the self-avoiding walk, percolation, orientedpercolation, the contact process, lattice trees and -animals, and the Ising model. This paper provides ageneralized 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 particularinterest are those spread-out models where the underlying step distribution has infinite variance, so-called 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 expansioncoefficients 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.sing this generalized framework, we do the analysis of the lace expansion in such a way that itholds for any model provided that the expansion has a specific form and certain bounds on the laceexpansion coefficients are satisfied (see Section 2). These bounds are proved to follow from a relatedrandom walk condition, which is relatively simple to verify.
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 . Wewill refer to the edges as bonds and to the vertices as sites . We assign each (undirected) bond { x, y } aweight 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 anarbitrary element from the Fourier dual of the discrete lattice, which is the torus [ − π, π ) d . The Fouriertransform of a summable function f : Z d → C is defined by ˆ f ( k ) = P x ∈ Z d f ( x ) e ik · x . D : 3 versions Let D denote a probability distribution on Z d that is symmetric under reflections in coordinate hyper-planes and rotations by π/
2. We refer to D as a step distribution, having in mind a random walkertaking independent steps distributed according to D . Without loss of generality we henceforth assumethat 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 resultsfor these versions, they actually hold more generally under a random walk condition formulated in As-sumption 2.1 below. The first version is the nearest-neighbor model , where D is the uniform distributionon the nearest neighbors, i.e., D ( x ) = 12 d {| x | =1 } , x ∈ Z d . (1.1)Here, and throughout the paper, we denote by | · | the Euclidian norm on Z d and E represents theindicator function of the event E . This nearest-neighbor version of D corresponds to the classical modelfor 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 thedefinitions, but later omit the subscript. In the finite-variance spread-out model we require D L tosatisfy the following conditions :(D1) There is an ε > X x ∈ Z d | x | ε D L ( x ) < ∞ . (D2) There is a constant C such that, for all L ≥ k D L k ∞ ≤ CL − d . (D3) There exist constants c , c > − ˆ D L ( k ) ≥ c L | k | if k k k ∞ ≤ L − , (1.2)1 − ˆ D L ( k ) > c if k k k ∞ ≥ L − , (1.3)1 − ˆ D L ( k ) < − c , k ∈ [ − π, π ) d . (1.4) These conditions coincide with Assumption D in [32]. xample. 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 byninety degrees. Assume that there is an integrable function H on R d with H ( te ) non-increasing in t ≥ e ∈ R d , such that h ( x ) ≤ H ( x ) for all x ∈ R d . Assume further that the (2 + ε )-thmoment of h exists for some ε >
0. The monotonicity and integrability hypotheses on H imply that P x h ( x/L ) < ∞ for all L , with x/L = ( x /L, . . . , x d /L ). Then D L ( x ) = h ( x/L ) P y ∈ Z d h ( y/L ) , x ∈ Z d , (1.5)obeys the conditions (D1)–(D3), whenever L is large enough (cf. [32, Appendix A]). For h ( x ) = { < k x k ∞ ≤ } we obtain the uniform spread-out model with D L ( x ) = 1(2 L + 1) d − { < k x k ∞ ≤ L } , x ∈ Z d . (1.6)In the spread-out power-law model we replace assumptions (D1) and (D3) by the condition thatthere exists an α > ′ ) all ε > X x ∈ Z d | x | α − ε D L ( x ) < ∞ ;(D3 ′ ) there exist constants c , c > − ˆ D L ( k ) ≥ c L α | k | α if k k k ∞ ≤ L − , (1.7)1 − ˆ D L ( k ) > c if k k k ∞ ≥ L − , (1.8)1 − ˆ D L ( k ) < − c , k ∈ [ − π, π ) d . (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 momentof 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 P 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. h ( x ) = ( | x | ∨ − d − α , (1.11)so that D L has the form D L ( x ) = ( | x/L | ∨ − d − α P y ∈ Z d ( | y/L | ∨ − d − α , x ∈ Z d . (1.12)Chen and Sakai [18, Prop. 1.1] showed that, analogously to the finite-variance spread-out model, thespread-out power-law model (1.12) satisfies conditions (D1 ′ )–(D3 ′ ).Note that the spread-out power-law model with parameter α > α ∧ α and 2 in the spread-out power-law case, and 2 in the nearest-neighborcase 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 thesupport 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 theIsing model. 3 .2.2 Self-avoiding walk For every lattice site x ∈ Z d , we denote by W n ( x ) = { ( w , . . . , w n ) | w = 0 , w n = x, w i ∈ Z d , ≤ i ≤ n − } (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 ∈ { , . . . , n } . We define c ( x ) = δ ,x and, for n ≥ c n ( x ) := X w ∈W n ( x ) n Y i =1 D ( w i − w i − ) { w is self-avoiding } . (1.14)where D is as in Section 1.2.1. 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 oth-erwise. Thus for the nearest-neighbor model, each nearest-neighbor bond is occupied with probability z/ (2 d ). The corresponding product measure is denoted by P z with corresponding expectation E z . Werequire z ∈ [0 , k D k − ∞ ] to ensure that zD ( x − y ) ≤
1. We write { x ↔ y } for the event that there existsa path of occupied bonds from x to y . When the event { x ↔ y } occurs we call the vertices x and y connected . For x ∈ Z d , the set C ( x ) := { y ∈ Z d | y ↔ x } of connected vertices is called the cluster of x . It is the size and geometry of these clusters that we are interested in. Due to the shift invariance ofthe 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 ≥ z , the probability that the size of thecluster C is infinite, θ ( z ) := P z ( |C| = ∞ ) , (1.15)is strictly greater than zero. Since z θ ( z ) is non-decreasing, there exists some critical value z c wherethis probability turns positive (see e.g. [24]). For the Ising model we consider the space {− , } 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 , andfor every spin configuration ϕ = { ϕ x | x ∈ Λ } ∈ {− , } Λ the energy given by the Hamiltonian H Λ ( ϕ ) = − X { x,y }∈ Λ × Λ J ( y − x ) ϕ x ϕ y , (1.16)where J and D are related via the identity D ( x ) = tanh( zJ ( x )) P y ∈ Z d tanh( zJ ( y )) , (1.17)and z is the inverse temperature. For example, in the nearest-neighbor case, D = J . For the Isingmodel, J is known as the spin-spin coupling . If J ≥ D ≥
0, as in the cases we consider)then the model is called ferromagnetic . 4 .2.5 Two-point function and susceptibility
We study self-avoiding walk, percolation and the Ising model in a unified way. For this, we need tointroduce some notation. We consider the function G z ( x ) for x ∈ Z d with G z ( x ) = ∞ X n =0 c n ( x ) z n (1.18)being the Green’s function for self-avoiding walk, while for percolation G z ( x ) = P z (0 ↔ x ) (1.19)being the probability of the event that there is a path consisting of occupied edges from 0 to x . For theIsing model, we consider the spin correlation G z as the thermodynamic limit G z ( x ) = lim Λ ր Z d P ϕ ∈{− , } Λ ϕ ϕ x exp( − z H Λ ( ϕ )) P ϕ ∈{− , } Λ exp( − z H Λ ( ϕ )) . (1.20)Here the limit is taken over any non-decreasing sequence of Λ’s converging to Z d . This limit exists andis 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 modelsdepending on the two points 0 and x .We further introduce the susceptibility as χ ( z ) := X x ∈ Z d G z ( x ) . (1.21)For percolation, the susceptibility is the expected cluster size χ ( z ) = E z |C| .We define z c , the critical value of z , as z c := sup { z | χ ( z ) < ∞} . (1.22)For self-avoiding walk, z c is the convergence radius of the power series (1.18). For percolation, z c ischaracterized by the explosion of the expected cluster size. Menshikov [35], as well as Aizenman andBarsky [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 exemplifiedin [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 M ( z, h ) = lim Λ ր Z d P ϕ ∈{− , } Λ ϕ exp {− z H Λ ( ϕ ) + h P y ∈ Λ ϕ y } P ϕ ∈{− , } Λ exp {− z H Λ ( ϕ ) + h P y ∈ Λ ϕ y } , (1.23)and write M ( z, + ) for the limit lim h ց M ( z, h ). The magnetization gives rise to another characteriza-tion of z c , namely z c = inf { z | M ( z, + ) > } . As proved by Aizenman, Barsky and Fern´andez [3], thisis equivalent to (1.22). 5 .2.6 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 howmodels behave at and nearby this critical value. We use the notion of critical exponents to describethis behavior. While the existence of these critical exponents is folklore, there is no general argumentproving this.We write f ( z ) ≍ g ( z ) if the ratio f ( z ) /g ( z ) is bounded away from 0 and infinity, for some appropriatelimit. For self-avoiding walk, we define the critical exponents γ S and η S by χ ( z ) ≍ ( z c − z ) − γ S as z ր z c , (1.24)ˆ G z c ( k ) ≍ | k | ( α ∧ − η S as k →
0. (1.25)For percolation we define the critical exponents γ P , β P , δ P and η P by χ ( z ) ≍ ( z c − z ) − γ P as z ր z c , (1.26) θ ( z ) ≍ ( z − z c ) β P as z ց z c , (1.27) P z c ( |C| ≥ n ) ≍ n /δ P as n → ∞ , (1.28)ˆ G z c ( k ) ≍ | k | ( α ∧ − η P as k →
0. (1.29)The exponent γ P describes the asymptotic behavior in the subcritical regime { z < z c } , β P describes thebehavior in the supercritical regime { z > z c } , and δ P and η P describe the behavior at criticality. Forthe Ising model, we consider the critical exponents γ I , β I , δ I , η I defined by χ ( z ) ≍ ( z c − z ) − γ I as z ր z c , (1.30) M ( z, ≍ ( z − z c ) β I as z ց z c , (1.31) M ( z c , h ) ≍ h /δ I as h ց
0, (1.32)ˆ G z c ( k ) ≍ | k | ( α ∧ − η I as k →
0. (1.33)For a discussion on the construction of ˆ G z c ( k ) we refer to Section 2.1 below.It is believed that critical exponents are universal , i.e., minor modifications of the model, like changesin 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 uppercritical dimension d c , such that the critical exponents take the same value for all d > d c . These valuesare the mean-field values of the critical exponents. For self-avoiding walk these are the values obtainedfor 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 obtainedfor percolation on an infinite regular tree, see [24, Section 10.1]. For the Ising model, these mean-fieldvalues are γ I = 1, β I = 1 / 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 theirmean-field values in sufficiently high dimensions for the nearest-neighbor version of D , or d exceedingsome critical dimension d c and L sufficiently large for the spread-out models, respectively. We introduce the (small) quantity β by β = K/d for the nearest-neighbor model ( K is a uniformconstant), or β = K L − d for the spread-out models ( K is a constant depending on d and α ). We make6his 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 percola-tion, and τ ( z ) = X y ∈ Z d tanh( zJ ( y )) (1.34)for the Ising model, cf. (1.17).Our main result is the following infrared behavior: Theorem 1.1 (Infrared bound) . Fix s = 2 for self-avoiding walk and the Ising model, and s = 3 forpercolation. Let d sufficiently large in the nearest-neighbor case (at least d > s ), or d > s and L sufficiently large in the finite-variance spread-out case, or d > ( α ∧ s and L sufficiently large in thespread-out power-law case. Then ˆ G z ( k ) = 1 + O ( β ) χ ( z ) − + τ ( z )[1 − ˆ D ( k )] (1.35) uniformly for z ∈ [0 , z c ) and k ∈ [ − π, π ) d . The infrared bound is well-known in several cases. Hara and Slade proved the infrared bound forthe nearest-neighbor case and the finite-variance spread-out case, for self-avoiding walk [27, 28] (seealso [34, Theorem 6.1.6]) as well as for percolation [26]. Fr¨ohlich, Simon and Spencer [22] proved theupper 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 ) − in (1.35), we obtain from Theorem 1.1 that (under the assumptionsformulated there) ˆ G z ( k ) ≤ O ( β ) τ ( z )[1 − ˆ D ( k )] (1.36)uniformly for z < z c .Note that the bound G z ( x ) − δ ,x ≤ τ ( z ) ( D ∗ G z ) ( x ) . (1.37)holds in all our three models: for self-avoiding walk this is obvious, for percolation it follows from theBK-inequality [11], and for the Ising model we use [38, (4.2)] in the infinite-volume limit. Thus for s = 2, B ( z ) := X x G z ( x ) ≤ X x τ ( z ) ( D ∗ G z ) ( x ) ≤ τ ( z ) Z [ − π,π ) d ˆ D ( k ) ˆ G z ( k ) d k (2 π ) d . (1.38)A combination of (1.36) and (1.38) gives rise to B ( z ) ≤ O (1) Z [ − π,π ) d ˆ D ( k ) [1 − ˆ D ( k )] d k (2 π ) d ≤ O ( β ) , (1.39)where we use that the integrated term is O ( β ) by Assumption 2.1 and Proposition 2.2 below. A similarcalculation gives the corresponding result for s = 3. More specifically, T ( z ) := X x,y G z (0 , x ) G z ( x, y ) G z ( y, ≤ O ( β ) when s = 3. (1.40)The bounds (1.38)–(1.40) hold uniformly for z < z c under the assumptions in Theorem 1.1. Note thatin (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 .7he two-point function G z ( x ) seen as a function of z (for fixed x ) is continuous. For self-avoidingwalk 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 functionwhich is continuous and non-decreasing in z , hence G z ( x ) is left-continuous (cf. [25, Appendix A]). Itfollows that (1.38)–(1.40) even hold at criticality, i.e. when z = z c . In particular, this implies the bubblecondition (i.e., B ( z c ) < ∞ ) or the triangle condition (i.e., T ( z c ) < ∞ ) for s = 2 or 3, respectively. Weformulate this fact as a corollary: Corollary 1.2 (Bubble/Triangle condition) . Under the assumptions in Theorem 1.1, B ( z c ) ≤ O ( β ) for s = 2 (self-avoiding walk and Ising model), and T ( z c ) ≤ O ( β ) for s = 3 (percolation). The bubble/triangle condition is important since it implies mean-field behavior of the model, whichis formulated in the next theorem. In fact, (1.35) extends to the critical case z = z c asˆ G z c ( k ) = 1 + O ( β )1 − ˆ D ( k ) , (1.41)and we refer to the discussion around (2.7) below for a construction of ˆ G z c ( k ) and a derivation of (1.41).We now use Theorem 1.1 to establish the existence of the formerly introduced critical exponents. Theorem 1.3 (Critical exponents) . (i) Self-avoiding walk.
Consider the self-avoiding walk model ( s = 2 ). Under the assumptions inTheorem 1.1, the critical exponent γ S = 1 for the self-avoiding walk exists.(ii) Percolation.
Consider the percolation model ( s = 3 ). Under the assumptions in Theorem 1.1,the critical exponents γ P = 1 , β P = 1 and δ P = 2 for percolation exist.(iii) Ising model.
Consider the Ising model ( s = 2 ). Under the assumptions in Theorem 1.1, thecritical exponents γ I = 1 , β I = 1 / and δ I = 3 for the Ising model exist.(iv) For all three models, under the assumptions in Theorem 1.1 and if − ˆ D ( k ) ≍ | k | α ∧ , then ˆ G z c ( k ) ≍ | k | α ∧ as k → , (1.42) 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 well-known in the literature. However, the mode of convergence required for the existence of the criticalexponents varies, and some derivations are stated only for finite range models. We therefore add a moredetailed discussion of the literature here.For self-avoiding walk, the existence (and the value) of the critical exponent γ S is based on theinequality z c z c − z ≤ χ ( z ) ≤ B ( z c ) (cid:18) z c z c − z + 1 (cid:19) . (1.43)Thus the bubble condition (1.38) is sufficient to prove that γ S exists and that γ S = 1. The inequality(1.43) is derived from a differential inequality in [41, Theorem 2.3], which was proved there for uniformspread-out models. The derivation still holds for infinite-range spread-out models due to the multiplica-tive structure of the weights of the self-avoiding walks in (1.14). A version of (1.43) appeared earlier in[15, (5.30)–(5.33)]. 8he derivation of the exponents γ P = 1, β P = 1 and δ P = 2 from the triangle condition is due toAizenman–Newman [8] and Barsky–Aizenman [10]. To apply these results in our settings, there aresome 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 conditionimplies γ I = 1 as long as | J | = P x J ( x ) < ∞ (which is equivalent to P x D ( x ) < ∞ ). Under thesame condition, Aizenman and Fern´andez [5] proved the existence and mean-field values of the criticalexponents β I and δ I .The statement in (iv) is an immediate consequence of (1.41). The lower bound in 1 − ˆ D ( k ) ≍ | k | α ∧ follows from (D3)/(D3’). The upper bound indeed holds for a number of examples, and in particularif D is chosen as in the nearest-neighbor model (1.1), the finite-variance spread-out model (1.6) or thespread-out power-law model (1.12) with α = 2, cf. [18, 32]. However, if D is chosen as in (1.12) with α = 2, then 1 − ˆ D ( k ) ≍ ( L | k | ) log( π/ ( L | k | )) , cf. [18, Prop. 1.1].The proof of Theorem 1.1, as well as the proof of Corollary 1.2, is given at the end of Section 2. There is numerous work on the application of the lace expansion, see the lecture notes by Slade [41] andreferences therein. We give more references below at places where we use lace expansion methodologyand 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 weaklyself-avoiding walk in dimension d ≥ m lattice sites along the coordinate axes with probabilityproportional to 1 /m converges to a Cauchy process (as for ordinary random walk with such stepdistribution). 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-outcase when D has exponential tails. The study of long-range percolation with power law spread-outbonds started in the 1980’s by considering the one-dimensional case [9, 36, 39]. These authors studythe case where occupation probabilities are given by (1.12) with α ∈ (0 ,
1] and prove criteria for theexistence of an infinite cluster. For example, Aizenman–Newman [9] show that if D ( x ) | x | → | x | → ∞ in one dimension, and D (1) is sufficiently large, then there exists a critical infinite cluster andhence 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 > α (and here α = 1). Berger [12]uses a renormalization argument to show that in dimension d = 1 , < 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 aninfinite critical cluster for d ≥ α ≥ d [12, Question 6.4] is answered negatively by the presentpaper for d > L sufficiently large.In a recent paper, Chen and Sakai [18] study oriented percolation in the spread-out power-lawcase. Using similar methods, they prove that the two-point function in oriented percolation obeys aninfrared bound if d > α ∧ Ising model in one dimension has been studied by Aizenman, Chayes, Chayes, andNewman [4]. Similar to the percolation result in [9], they prove that in the one-dimensional case where D ( x ) | x | → | x | → ∞ , the spontaneous magnetization M ( z, z c .The infrared bound for the Ising model was proved in [22] for d > α ∧ reflection positivity (RP) property. The class of models satisfying (RP) includes the nearest-neighbormodel (where D ( x ) = (2 d ) − {| x | =1 } ), exponential decaying potentials (where D ( x ) ∝ exp {− µ k x k } for µ > D ( x ) ∝ | x | − s for s > η (when it exists) is nonnegative. Our approach using the lace expansion does not requirereflection positivity, it is much more universal in the choice of D (cf. Section 1.2.1), and also gives amatching lower bound in (1.35), yielding η = 0. On the other hand, our approach requires that thedimension d or the spread-out parameter L are sufficiently large, a limitation that one may not expectto reflect the physics. The literature for the long-range Ising model in higher dimensions based on (RP)arguments is summarized by Aizenman and Fern´andez [6], who also identify 2( α ∧
2) as upper criticaldimension. Given (1.42) it is folklore that G z c ( x ) ≍ | x | − d +( α ∧ (1.44)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-avoidingwalk and percolation, Hara [25] proved it in the nearest-neighbor setting, and Sakai [38] proved it for theIsing model in finite-range spread-out and nearest-neighbor settings. We discuss the critical two-pointfunction G z c ( x ) at the end of Sect. 2.1. In order to study the various models in a unified way, we use this section to set up a generalizedframework. We make two assumptions in terms of the general framework, and use the subsequent twosections to show that our models actually satisfy these assumptions. We then prove the results withinthe abstract setting, based on the two assumptions made.
Given a step distribution D , we consider the random walk two-point function or Green’s function of therandom walk defined by C z ( x ) = ∞ X n =0 D ∗ n ( x ) z n , (2.1)where D ∗ n is the n -fold convolution of D and D ∗ ( x ) z = δ x, . We write δ for the Kronecker deltafunction. By conditioning on the first step we obtain C z ( x ) = δ ,x + z ( D ∗ C z ) ( x ) . (2.2)Taking the Fourier transform and solving for ˆ C z ( k ) yieldsˆ C z ( k ) = 11 − z ˆ D ( k ) , z < . (2.3)Next we consider G z ( x ) defined in (1.18)–(1.20). For each of the three models, i.e., for self-avoidingwalk, percolation and the Ising model, we use the lace expansion to obtain an expansion formula of theform G z ( x ) = δ ,x + τ ( z ) ( D ∗ G z ) ( x ) + ( G z ∗ Φ z ) ( x ) + Ψ z ( x ) . (2.4)The coefficients Φ z ( x ) and Ψ z ( x ) depend on the model, but above their respective upper critical di-mension they obey similar bounds. Assuming the existence of ˆΦ z ( k ) and ˆΨ z ( k ), Fourier transformationyields ˆ G z ( k ) = 1 + ˆΨ z ( k )1 − τ ( z ) ˆ D ( k ) − ˆΦ z ( k ) , z < z c . (2.5) There is a typo in [6], the value of δ in [6, (1.2)] should be 3. z and Ψ z , and citebounds on them from [14, 38, 41]. We will see that, for z = 0, ˆΨ ( k ) ≡ ( k ) ≡ τ ( z ) = z for self-avoiding walk and percolation, and τ ( z ) = P y ∈ Z d tanh( zJ ( y )) for theIsing model, see Sect. 1.3.For the critical case (i.e., z = z c ) we have1 ≤ τ ( z c ) ≤ O ( β ) , (2.6)where the lower bound is a consequence of (1.37), and the upper bound emerges from (2.18) and (2.29)below. The function G z c ( x ) = lim z ր z c G z ( x ) is not in ℓ ( Z d ), hence the Fourier transform does notexist. However, diagrammatic bounds of the lace expansion coefficients (Prop. 2.5) and the dominatedconvergence theorem guarantee the absolute convergence of the various sums involved defining ˆΨ z ( k )and ˆΦ z ( k ), which shows that the critical quantities ˆΨ z c ( k ) and ˆΦ z c ( k ) are well-defined. This justifiesthe introduction of ˆ G z c ( k ) as a solution to (2.5) with z = z c . Note that we do not assume any continuityof z ˆΨ z ( k ) and z ˆΦ z ( k ) to do this. Nevertheless, we can extend (1.35) to the critical case z = z c ,and further use (2.6) to obtain ˆ G z c ( k ) = 1 + O ( β )1 − ˆ D ( k ) . (2.7)An issue of interest is the (left-) continuity of ˆ G z ( k ) at z = z c . In particular, the identity G z c ( x ) = Z [ − π,π ) d e − ik · x ˆ G z c ( k ) d k (2 π ) d , x ∈ Z d , (2.8)would follow from the the fact that ˆΨ z ( k ) and ˆΦ z ( k ) are left-continuous at z = z c , as explained by Hara[25, Appendix A]. The left-continuity of ˆΨ z ( k ) and ˆΦ z ( k ) at z = z c indeed holds for self-avoiding walk(by Abel’s Theorem) and for percolation (by [25, Lemma A.1]), but a proof for the Ising model is notknown. Recall that the model parameter s is 2 for self-avoiding walk or Ising model, and 3 for percolation. Wenow make an assumption on the step distribution D . Assumption 2.1 (Random walk s -condition) . There exists β > sufficiently small such that sup x ∈ Z d D ( x ) ≤ β (2.9) and Z [ − π,π ) d ˆ D ( k ) [1 − ˆ D ( k )] s d k (2 π ) d ≤ β. (2.10) Remark:
The specific amount of smallness required in (2.9)–(2.10) will be specified in the proofs inSection 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 ( D ∗ C ∗ D ∗ C )(0) ≤ β. (2.11)In other words, we have an (ordinary) random walk from 0 to x of at least one step, and a secondwalk from x to 0 and subsequently sum over all x . Correspondingly, for s = 3, we obtain the x -spacerepresentation ( C ∗ D ∗ C ∗ D ∗ C )(0) ≤ β, (2.12)and refer to (2.10) as the random walk triangle condition. See the graphical representation in Figure 1.11 (y)0 (x)(x) Figure 1: Graphical representation of the random walk bubble diagram in (2.11) and the random walktriangle diagram in (2.12). A line between two points, say x and y , represents the two-point function C ( 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 )( 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 largein the nearest-neighbor model (at least d > s ) or d > d c = s ( α ∧ and L is sufficiently large in thespread-out models. More specifically, the assumption holds with β = O ( d − ) in the nearest-neighborcase, and β = O ( L − d ) in the spread-out cases. We prove Proposition 2.2 in Section 3. We shall prove the following generalized version of Theorem1.1. 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. IfAssumption 2.1 is satisfied for β sufficiently small, then (1.35) holds uniformly for z ∈ [0 , z c ) and k ∈ [ − π, π ) d . We remark that Theorem 1.3 generalizes in the same way.
We introduce the quantity λ z := 1 − G z (0) = 1 − χ ( z ) ∈ [0 , . (2.13)Then λ z satisfies the equality ˆ G z (0) = ˆ C λ z (0) . (2.14)The idea of the proof of Theorem 2.3 is motivated by the intuition that ˆ G z ( k ) and ˆ C λ z ( k ) are comparablein size and, moreover, the discretized second derivative∆ k ˆ G z ( l ) := ˆ G z ( l − k ) + ˆ G z ( l + k ) − G ( l ) (2.15)is bounded by U λ z ( k, l ) := 200 ˆ C λ z ( k ) − n ˆ C λ z ( l − k ) ˆ C λ z ( l ) + ˆ C λ z ( l ) ˆ C λ z ( l + k ) + ˆ C λ z ( l − k ) ˆ C λ z ( l + k ) o . (2.16)More precisely, we will show that the function f : [0 , z c ) → R , defined by f := f ∨ f ∨ f (2.17)with f ( z ) := τ ( z ) , f ( z ) := sup k ∈ [ − π,π ) d ˆ G z ( k )ˆ C λ z ( k ) , (2.18)12nd f ( z ) := sup k,l ∈ [ − π,π ) d | ∆ k ˆ G z ( l ) | U λ z ( k, l ) , (2.19)is small, given that β in Assumption 2.1 is sufficiently small. To make this rigorous, we need thefollowing assumption: Assumption 2.4 (Bounds on the lace expansion coefficients) . If, for some
K > , the inequality f ( z ) ≤ K holds uniformly for z ∈ (0 , z c ) , then there exists a constant c K > such that, for all k ∈ [ − π, π ) d , (cid:12)(cid:12)(cid:12) ˆΨ z ( k ) (cid:12)(cid:12)(cid:12) ≤ c K β, (cid:12)(cid:12)(cid:12) ˆΦ z ( k ) (cid:12)(cid:12)(cid:12) ≤ c K β (2.20) and X x [1 − cos( k · x )] | Ψ z ( x ) | ≤ c K β ˆ C λ z ( k ) − , X x [1 − cos( k · x )] | Φ z ( x ) | ≤ τ ( z ) c K β ˆ C λ z ( k ) − (2.21) 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 Theorem1.1):
Proof of Theorem 2.3 subject to (2.20)–(2.21).
Let m z = 1 − τ ( z ) − ˆΦ z (0) . (2.22)Then, ˆ G z ( k ) = 1 + ˆΨ z ( k )1 − τ ( z ) ˆ D ( k ) − ˆΦ z ( k ) = 1 + ˆΨ z ( k ) m z + τ ( z )[1 − ˆ D ( k )] + [ ˆΦ z (0) − ˆΦ z ( k )] . (2.23)By the first inequality in (2.20) and the second in (2.21) in Assumption 2.4,ˆ G z ( k ) = 1 + O ( β ) m z + τ ( z ) [1 − ˆ D ( k )] + τ ( z ) O ( β ) ˆ C λ z ( k ) − . (2.24)Evaluating (2.23) for k = 0 yields χ ( z ) = ˆ G z (0) = 1 + ˆΨ z (0) m z , (2.25)and the first inequality in (2.20) implies m z = (1 + O ( β )) χ ( z ) − . (2.26)Furthermore, by (2.3) and (2.13),ˆ C λ z ( k ) − = 1 − λ z ˆ D ( k ) = 1 − ˆ D ( k ) + χ ( z ) − ˆ D ( k ) . (2.27)A combination of (2.24), (2.26), (2.27) and the bounds | ˆ D ( k ) | ≤ τ ( z ) ≤ O (1) leads toˆ G z ( k ) = 1 + O ( β )(1 + O ( β )) χ ( z ) − + τ ( z ) (1 + O ( β )) [1 − ˆ D ( k )] , (2.28)which implies (1.35).We proceed by validating (2.20)–(2.21). First we realize that Assumption 2.4 indeed holds for themodels under consideration: 13 roposition 2.5. Under the assumptions in Theorem 1.1, Assumption 2.4 holds for self-avoiding walk,percolation and the Ising model.
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 thediagrammatic bounds proved in these papers, and relate them to our version of Φ z and Ψ z , thusproving Proposition 2.5 using [14, 38, 41]. 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 ∈ { , , . . . } ,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 > . Assume further that χ ′ ( z ) ≤ O ( χ ( z ) ) , z ∈ [0 , z c ) . Then f ( z ) ≤ O ( β ) (2.29) uniformly for z < z c . The assumption χ ′ ( z ) ≤ const χ ( z ) in Proposition 2.6 can be replaced by assuming that f iscontinuous on [0 , z c ), cf. Lemma 5.3 below. It is known as a mean-field bound , and a proof of it can befound in [41, Theorem 2.3] for self-avoiding walk, and in [41, Prop. 9.2] for percolation. For the Isingmodel, this mean-field bound is a consequence of the Lebowitz inequality [33].In order for Theorem 2.3 (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. Propositions2.2 and 2.5 validate Assumptions 2.1 and 2.4. With these assumptions, the prerequisites of Proposition2.6 are satisfied and (2.29) holds for β sufficiently small by Proposition 2.2. The latter can be achievedby taking d or L large enough. Then we again use Assumption 2.4 to obtain (2.20)–(2.21), thus proving(1.35).The remainder of the paper is organized as follows. In Section 3 we prove Proposition 2.2 by showingthat 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 explainin Section 4 how to obtain the statement of Proposition 2.5 from the diagrammatic bounds in [41] forself-avoiding walk, [14] for percolation, and [38] for the Ising model. Finally, the proof of Proposition 2.6is contained in the last Section 5, and this completes the proof of Theorem 2.3 (and hence of Theorem1.1 and Corollary 1.2). Appendix A contains a derivation of the existence and the mean-field valuesof the critical exponents γ P and δ P for percolation. In Appendix B we show how the bounds on thelace expansion in Assumption 2.4 for the Ising model can be obtained from the diagrammatic boundsin [38]. Our account in Appendix B follows the proof of [38, Prop. 3.2], but with a modified bootstraphypothesis. In this section we prove Proposition 2.2. The estimates below are contained in [14, Sect. 2.2.2], wherefinite tori are considered. Restriction to the infinite lattice gives rise to a noteworthy simplification,which we shall present in the following.
Proof of Proposition 2.2 for the nearest-neighbor model.
We follow [14, Sect. 2.2.2]. Since k D k ∞ =(2 d ) − , the bound (2.9) is satisfied for d sufficiently large, and it remains to prove (2.10).14y the symmetry of D we haveˆ D ( k ) = X x ∈ Z d D ( x ) cos( k · x ) = 1 d d X j =1 cos( k j ) , k = ( k , . . . , k d ) ∈ [ − π, π ) d . (3.1)Since 1 − cos t ≥ π − t for | t | ≤ π , this implies the infrared bound1 − ˆ D ( k ) ≥ π | k | d . (3.2)The Cauchy-Schwarz inequality yields Z [ − π,π ) d ˆ D ( k ) [1 − ˆ D ( k )] s d k (2 π ) d ≤ Z [ − π,π ) d ˆ D ( k ) d k (2 π ) d ! / Z [ − π,π ) d − ˆ D ( k )] s d k (2 π ) d ! / (3.3)First we show that the first term on the right hand side of (3.3) is small if d is large. Note that R [ − π,π ) d ˆ D ( k ) (2 π ) − d d k = D ∗ (0) is the probability that a nearest-neighbor random walk returns to itsstarting point after the fourth step. This is bounded from above by c (2 d ) − with c being a well-chosenconstant, because the first two steps must be compensated by the last two. Finally, the square rootyields the upper bound O ( d − ).It remains to show that the second term on the right of (3.3) is bounded uniformly in d . The infraredbound (3.2) gives Z [ − π,π ) d − ˆ D ( k )] s d k (2 π ) d ≤ π s s Z [ − π,π ) d d s | k | s d k (2 π ) d . (3.4)The right hand side of (3.4) is finite if d > s . For A > m > A m = 1Γ( m ) Z ∞ t m − e − tA d t. (3.5)Applying this with A = | k | /d and m = 2 s yields1Γ(2 s ) π s s Z ∞ t s − Z π − π (cid:0) e − tθ (cid:1) /d d θ π ! d d t (3.6)as an upper bound for (3.4). This is non-increasing in d , because k f k p ≤ k f k q for 0 < p ≤ q ≤ ∞ on aprobability space by Lyapunov’s inequality. Proof of Proposition 2.2 for the spread-out models.
We again follow [14, Sect. 2.2.2]. Obviously (2.9) isimplied by condition (D2)/(D2 ′ ) for sufficiently large L , hence it remains to prove (2.10).The power-law spread-out model with α > ε <α −
2. Note further that (D3) and (D3 ′ ) agree when the exponent in the first inequality is taken α ∧ k k k ∞ ≤ L − and k k k ∞ > L − . By (1.2), (1.7) and the boundˆ D ( k ) ≤
1, the corresponding contributions to the integral are Z k : k k k ∞ ≤ L − ˆ D ( k ) [1 − ˆ D ( k )] s d k (2 π ) d ≤ c s L ( α ∧ s Z k : k k k ∞ ≤ L − | k | ( α ∧ s d k (2 π ) d ≤ C d,c L − d (3.7) The H¨older inequality gives better bounds here. In particular, it requires d > s only, cf. (2.19) in [14]. d > ( α ∧ s , where C d,c is a constant depending (only) on d and c , and by (1.3), (1.8), Z k : k k k ∞ >L − ˆ D ( k ) [1 − ˆ D ( k )] s d k (2 π ) d ≤ c − s Z k : k k k ∞ >L − ˆ D ( k ) d k (2 π ) d ≤ const L − d , (3.8)for some positive constant. In the last step we used assumption (D2) / (D2 ′ ) to see that Z k ∈ [ − π,π ) d ˆ D ( k ) d k (2 π ) d = ( D ∗ D )(0) = X y ∈ Z d D ( y ) ≤ X y ∈ Z d D ( y ) k D k ∞ = k D k ∞ ≤ const L − d . (3.9) In this section, we discuss the lace expansion which obtains an expansion of the two-point function ofthe form G z ( x ) = δ ,x + τ ( z ) ( D ∗ G z ) ( x ) + ( G z ∗ Φ z ) ( x ) + Ψ z ( x ) , cf. (2.4). The key point is to identify the lace-expansion coefficients Φ z and Ψ z in a way that allowsfor 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 self-avoiding walk, in [14] for percolation and in [38] for the Ising model. The lace expansion for self-avoiding walks was first presented by Brydges and Spencer [16]. They providean algebraic expansion using graphs. A special class of graphs that play an important role here, thelaces, gave the lace expansion its name. An alternative approach is based on an inclusion-exclusionargument, 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. Forexample, in [31, Sect. 2.2.1] it is shown that c n +1 ( x ) = ( D ∗ c n )( x ) + n +1 X m =2 ( π m ∗ c n +1 − m ) ( x ) (4.1)for suitable π m ( x ). We multiply (4.1) by z n +1 and sum over n ≥
0. By letting Π z ( x ) = P ∞ m =2 π m ( x ) z m and recalling G z ( x ) = P ∞ n =0 c n ( x ) z n this yields G z ( x ) = δ ,x + z ( D ∗ G z )( x ) + ( G z ∗ Π z )( x ) , (4.2)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 = β ( K ) , such that the followingholds: If Assumption 2.1 holds for some β ≤ β , then X x ∈ Z d | Π z ( x ) | ≤ c K β, (4.3) X x ∈ Z d [1 − cos( k · x )] | Π z ( x ) | ≤ c K β ˆ C λ z ( k ) − , k ∈ [ − π, π ) d . (4.4)16he 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, theproof in [41] can be modified to obtain X x ∈ Z d [1 − cos( k · x )] | Π z ( x ) | ≤ z c K β ˆ C λ z ( k ) − , k ∈ [ − π, π ) d . (4.5)instead of (4.4). This is achieved by leaving the factor z in [41, (5.42) and (5.43)] explicit (rather thenbounding above by K ).We choose τ ( z ) = z , Φ z ( x ) = Π z ( x ) and Ψ z ( x ) = 0, which makes (4.2) equivalent to (2.4). HenceProp. 4.1 along with (4.5) is sufficient to prove Proposition 2.5 for self-avoiding walk. The lace expansion for percolation was first derived in [26]. It is based on an inclusion-exclusionargument, and holds quite generally for any connected graph, finite or infinite. The graph does noteven need to be transitive or regular.In [14, Sect. 3.2], the identity G z ( x ) = δ ,x + z ( D ∗ G z ) ( x ) + z (Π M ∗ D ∗ G z ) ( x ) + Π M ( x ) + R M ( x ) (4.6)is derived for M = 0 , , , . . . . 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 expansionconverges, one has lim M →∞ X x | R M ( x ) | = 0 . (4.7)The subscript M denotes the level to which the (inclusion-exclusion) expansion is carried out, and weshall later fix M so large that (4.12) and (4.13) below are satisfied for K = 4. The equality (4.6) isequivalent to (2.4) if we let τ ( z ) = z , andΦ z ( x ) = z ( D ∗ Π M )( x ) , x ∈ Z d , (4.8)Ψ z ( x ) = Π M ( x ) + R M ( x ) , x ∈ Z d . (4.9)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 = β ( K ) , such that the following holds: IfAssumption 2.1 holds for some β ≤ β , then for all M = 0 , , , . . . , X x | Π M ( x ) | ≤ c K β, (4.10) X x [1 − cos( k · x )] | Π M ( x ) | ≤ c K β ˆ C λ z ( k ) − , (4.11) and for M sufficiently large (depending on K and z ), X x | R M ( x ) | ≤ β, (4.12) X x [1 − cos( k · x )] | R M ( x ) | ≤ β ˆ C λ z ( k ) − . (4.13)17n fact, the bounds in Proposition 4.2 are not exactly as phrased in [14]. In the following we explainhow the proof of [14, Prop. 5.2] can be modified to obtain Prop. 4.2. There are two differences toconsider. First, in the definition of f there is a factor 16 in the denominator, whereas we have a factor200, cf. (2.19). This can be controlled easily by changing the factor appropriately throughout the proofof [14, Prop. 5.2]. This changes the specific value of c K , but the statement of [14, Prop. 5.2] remainsunchanged. The second (and more important) issue is the replacement of 1 − ˆ D ( k ) = ˆ C ( k ) − in [14,Prop. 5.2] by 1 − λ z ˆ D ( k ) = ˆ C λ z ( k ) − in Prop. 4.2. We need to do this replacement to achieve continuityof the function f . Wherever the bound on f 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 − ˆ D ( k )] by ˆ C λ z ( k ) − .Other occurrences of [1 − ˆ D ( k )], as in [14, (5.75)] and [14, (5.91)], can be treated with the bound0 ≤ − ˆ D ( k ) ≤ C λ z ( k ) − , k ∈ [ − π, π ) d , (4.14)which itself is a consequence of0 ≤ ˆ C λ z ( k ) [1 − ˆ D ( k )] = 1 + λ z − − λ z ˆ D ( k ) ˆ D ( k ) ≤ . (4.15)Again, this increases the value of the constant c K , but leaves the statement of [14, Prop. 5.2] otherwiseunchanged.For a sketch of the argument of how f ( z ) ≤ K actually implies (4.10)–(4.13) we refer to [37, Sect.3.2]. 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) fol-low directly from Proposition 4.2 if M is chosen so large that (4.12)–(4.13) is satisfied.For the bounds on Φ z ( x ) = z ( D ∗ Π M )( x ) we use the estimate[1 − cos( t + t )] ≤ − cos t ] + [1 − cos t ]) , t , t ∈ R , (4.16)(see [14, (4.51)]) to obtain X x [1 − cos( k · x )] | Φ z ( x ) | ≤ X x z X y (cid:0) [1 − cos( k · y )] + [1 − cos( k · ( x − y ))] (cid:1) D ( y ) | Π M ( x − y ) |≤ z X x [1 − ˆ D ( k )] | Π M ( x − y ) | +5 z X x [1 − cos( k · ( x − y ))] | Π M ( x − y ) |≤ z (cid:16) c K β ˆ C λ z ( k ) − + c K β ˆ C λ z ( k ) − (cid:17) . (4.17)by (4.10)–(4.11) and (4.14). The lace expansion for the Ising model has been established recently by Sakai [38]. It is similar in spiritto 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 thatshowing some similarities to a percolation configuration. The lace expansion is then performed usingideas 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 G Λ z ( x ) = δ ,x + τ ( z ) (cid:0) D ∗ G Λ z (cid:1) ( x ) + τ ( z ) (cid:0) D ∗ Π Λ M ∗ G Λ z (cid:1) ( x ) + Π Λ M ( x ) + R Λ M ( x ) , (4.18)18here 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( − 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-volumetwo-point function. This is equivalent to (2.4) if we letΦ Λ z ( x ) = τ ( z )( D ∗ Π Λ M )( x ) , x ∈ Z d , (4.19)Ψ Λ z ( x ) = Π Λ M ( x ) + R Λ M ( x ) , x ∈ Z d , (4.20)then choose M so large that (4.23) and (4.24) below are satisfied for a certain K , say K = 4, andsubsequently taking the thermodynamic limit Λ ր Z d . Note that, if comparing (4.18) to [38, (1.11)], weexplicitly extract the δ ,x -term from the Π-term in [38], i.e., Π ( M ) p ;Λ ( x ) in [38] corresponds to Π Λ M ( x ) + δ ,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 = β ( K ) , such that the followingholds: If Assumption 2.1 holds for some β ≤ β , then for all M = 0 , , , . . . , X x | Π Λ M ( x ) | ≤ c K β, (4.21) X x [1 − cos( k · x )] | Π Λ M ( x ) | ≤ c K β ˆ C λ z ( k ) − , (4.22) and for M sufficiently large (depending on K and z ), X x | R Λ M ( x ) | ≤ β, (4.23) X x [1 − cos( k · x )] | R Λ M ( x ) | ≤ β ˆ C λ z ( k ) − . (4.24) These bounds hold uniformly in Λ . Since the bootstrapping hypothesis used in Section 5 in this paper is different from that in [38], itis not so obvious how Prop. 4.3 follows from the results in [38]. In Appendix B we explain how thestatement 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 ofProp. 4.2. We refrain from repeating the argument.
In this section we prove Proposition 2.6 and, by doing so, complete the proof of Theorem 1.1. Theproof is based on the following lemma:
Lemma 5.1 (The bootstrap / forbidden region argument) . Let f be a continuous function on theinterval [0 , z c ) , and assume that f (0) ≤ . Suppose for each z ∈ (0 , z c ) that if f ( z ) ≤ , then in fact f ( z ) ≤ . Then f ( z ) ≤ for all z ∈ [0 , z c ) .Proof. This is a straightforward application of the intermediate value theorem for continuous functions,see also [41, Lemma 5.9]. 19he bootstrap argument in Lemma 5.1 is often used in lace expansion, see e.g. [34, Section 6.1]. Analternative approach that involves an induction argument has been applied in [32], see also the lecturenotes by van der Hofstad [31].In the remainder of the section, we prove that the function f defined in (2.17) obeys the prerequisitesof Lemma 5.1. We therefore have to show that f (0) ≤
3, that f is continuous on [0 , z c ), and that f ( z ) ≤ f ( z ) ≤ z ∈ (0 , z c ). The latter is referred to as the improvement of the bounds .Let us first check that f (0) ≤
3. Clearly, f (0) = 0. Note that ˆΨ ( k ) ≡ ( k ) ≡
0. This leadsto ˆ G ( k ) ≡ λ = 0, hence f (0) = 1 and f (0) = 0.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 offunctions on an interval [ t , t ] , i.e., for every given ε > , there is a δ > such that | f α ( s ) − f α ( t ) | < ε whenever | s − t | < δ , uniformly in α ∈ A . Furthermore, suppose that sup α ∈ A f α ( t ) < ∞ for each t ∈ [ t , t ] . Then t sup α ∈ A f α ( t ) is continuous on [ t , t ] . A proof of this standard result can be found e.g. in [41, Lemma 5.12].
Lemma 5.3 (Continuity) . Assume that, for z ∈ (0 , z c ) , χ ′ ( z ) ≤ cχ ( z ) for some constant c . Then, thefunction f defined in (2.17) is continuous on (0 , z c ) .Proof. It is sufficient to show that f , f and f are continuous. The continuity of f is obvious. Weshow that f and f are continuous on the closed interval [0 , z c − ε ] for any ε > z and bound it uniformly in k on [0 , z c − ε ].We do f first. To this end, we consider the derivativedd z ˆ G z ( k )ˆ C λ z ( k ) = 1ˆ C λ z ( k ) " ˆ C λ z ( k ) d ˆ G z ( k )d z − ˆ G z ( k ) d ˆ C λ ( k )d λ (cid:12)(cid:12)(cid:12)(cid:12) λ = λ z d λ z d z . (5.1)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 seethat 12 ≤ − λ z ˆ D ( k ) = ˆ C λ z ( k ) ≤ ˆ C λ z (0) = χ ( z ) . (5.2)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 | ˆ G z ( k ) | ≤ | ˆ G z (0) | = χ ( z ) ≤ χ ( z c − ε ) , (5.3)For the derivative of the two-point function, we bound (cid:12)(cid:12)(cid:12)(cid:12) dd z ˆ G z ( k ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X x e ik · x dd z G z ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X x dd z G z ( x ) = dd z X x G z ( x ) = χ ′ ( z ) , (5.4)where the exchange in the order of sum and derivative is validated by the fact that both P x e ik · x G z ( x )and P x G z ( x ) are uniformly convergent series of functions. By the assumed mean-field bound χ ′ ( z ) ≤ cχ ( z ) , (5.4) is bounded above by cχ ( z c − ε ) .Moreover, we obtain from (2.3) that | d ˆ C λ ( k ) / d λ | ≤ ˆ C λ ( k ) , and, for λ = λ z , this is in turn boundedby χ ( z c − ε ) , cf. (5.2). Finally, | d λ z / d z | = χ ′ ( z ) /χ ( z ) ≤ c by (2.13) and our assumption.We treat f in exactly the same way as f , and omit the details here.20 .2 Improvement of the bounds The following lemma covers the remaining prerequisite of Lemma 5.1 and thus proves the final ingredientneeded for the proof of Proposition 2.6.
Lemma 5.4 (Improvement of the bounds) . If the assumptions of Proposition 2.6 are satisfied for somesufficiently small β , and if f ( z ) ≤ , then there exists a constant c > such that f ( z ) ≤ cβ for all z ∈ (0 , z c ) . In particular, if β is small enough, then f ( z ) ≤ . The following lemma will help us for the improvement of the bound on f . Lemma 5.5 (Slade [41]) . Suppose that a ( x ) = a ( − x ) for all x ∈ Z d , and let ˆ A ( k ) = 11 − ˆ a ( k ) . (5.5) Then, for all k, l ∈ [ − π, π ) d , (cid:12)(cid:12)(cid:12) ∆ k ˆ A ( l ) (cid:12)(cid:12)(cid:12) ≤ (cid:16) ˆ A ( l − k ) + ˆ A ( l + k ) (cid:17) ˆ A ( l ) (cid:16)c | a | (0) − c | a | ( k ) (cid:17) (5.6)+ 8 ˆ A ( l − k ) ˆ A ( l ) ˆ A ( l + k ) (cid:16)c | a | (0) − c | a | ( l ) (cid:17) (cid:16)c | a | (0) − c | a | ( k ) (cid:17) . By c | a | we denote the Fourier transform of the absolute value of a . The proof of Lemma 5.5 usesseveral bounds on trigonometric quantities, and can be found in [41, Lemma 5.7]. Proof of Lemma 5.4.
Fix z ∈ (0 , z c ) arbitrarily and assume f ( z ) ≤
4. Our general strategy will be toshow that f i for i = 1 , , β ) and thus, by taking β small, f ( z ) ≤ f is easy. First note that λ z = 1 − χ ( z ) − ≤
1. Using (2.13) along with (2.22)–(2.25)and Proposition 2.5 (with K = 4) we obtain f ( z ) = λ z (cid:16) z (0) (cid:17) − ˆΦ z (0) − ˆΨ z (0) ≤ λ z (cid:16) | ˆΨ z (0) | (cid:17) + | ˆΦ z (0) | + | ˆΨ z (0) |≤ c β. (5.7)The bound on f is slightly more involved. We write ˆ G z = ˆ N / ˆ F , withˆ N ( k ) = 1 + ˆΨ z ( k )1 + ˆΨ z (0) , ˆ F ( k ) = 1 − τ ( z ) ˆ D ( k ) − ˆΦ z ( k )1 + ˆΨ z (0) . (5.8)Recall from (2.3) that ˆ C λ z ( k ) = [1 − λ z ˆ D ( k )] − and, by (2.5) and (2.13), λ z = 1 − − τ ( z ) − ˆΦ z (0)1 + ˆΨ z (0) . (5.9)This yields ˆ G z ( k )ˆ C λ z ( k ) = ˆ N ( k ) + ˆ G z ( k ) h − λ z ˆ D ( k ) − ˆ F ( k ) i , (5.10)where 1 − λ z ˆ D ( k ) − ˆ F ( k ) = [1 − ˆ D ( k )] ˆΨ z (0) + [ ˆΦ z ( k ) − ˆΦ z (0)] ˆ D ( k ) + [1 − ˆ D ( k )] ˆΦ z ( k )1 + ˆΨ z (0) .
21y taking c β ≤ /
2, we obtain the bound1 + ℓc β − c β ≤ ℓ + 2) c β, ℓ = 0 , , , . . . , (5.11)which we use frequently below. For example, together with Assumption 2.4, it enables us to bound (cid:12)(cid:12)(cid:12) ˆ N ( k ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ( k )1 + ˆΨ z (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ˆΨ z ( k ) | − | ˆΨ z (0) | ≤ c β. Together with (4.14) we obtain in the same fashion that (cid:12)(cid:12)(cid:12) − λ z ˆ D ( k ) − ˆ F ( k ) (cid:12)(cid:12)(cid:12) ≤ [1 − ˆ D ( k )] | ˆΨ z (0) | + | ˆΦ z ( k ) − ˆΦ z (0) | + [1 − ˆ D ( k )] | ˆΦ z ( k ) | − | ˆΨ z (0) |≤ c β [1 − ˆ D ( k )] + c β ˆ C λ z ( k ) − − c β ≤ c β ˆ C λ z ( k ) − By our assumption that ˆ G z ( k ) ≤ C λ z ( k ) (which follows from f ( z ) ≤
4) and the above inequalities, wecan bound (5.10) from above by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ G z ( k )ˆ C λ z ( k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c β + 4 · c β (cid:12)(cid:12)(cid:12) ˆ C λ z ( k ) ˆ C λ z ( k ) − (cid:12)(cid:12)(cid:12) = 1 + 52 c β. (5.12)for every k ∈ [ − π, π ) d . This proves the bound on f .It remains to show the bound on f . In the following, we write K for a positive constant, whosevalue may change from line to line. Furthermore, we writeˆ G z ( k ) = ˆ b ( k )1 − ˆ a ( k ) , where ˆ b ( k ) = 1 + ˆΨ z ( k ), ˆ a ( k ) = τ ( z ) ˆ D ( k ) + ˆΦ z ( k ). (5.13)A straightforward calculation (see also [18, (4.18)]) shows that∆ k ˆ G z ( l ) = ∆ k ˆ b ( l )1 − ˆ a ( l ) + X σ ∈{ , − } (cid:0) ˆ a ( l + σk ) − ˆ a ( l ) (cid:1) (cid:0) ˆ b ( l + σk ) − ˆ b ( l ) (cid:1) (1 − ˆ a ( l )) (1 − ˆ a ( l + σk )) + ˆ b ( l ) ∆ k (cid:20) − ˆ a ( l ) (cid:21) . (5.14)We now bound all three summands in (5.14), and start with the first one: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ k ˆ b ( l )1 − ˆ a ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ k ˆ b ( l )ˆ b ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˆ G z ( l ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ k ˆΨ z ( l )1 + ˆΨ z ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˆ G z ( l ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ∆ k ˆΨ z ( l ) (cid:12)(cid:12)(cid:12) Kβ ) ˆ C λ z ( l ) , (5.15)where the last bound uses (2.21) to bound the denominator, and (5.12). A basic calculation shows thatany function g : Z d → R with g ( x ) = g ( − x ) satisfies | ∆ k ˆ g ( l ) | ≤ X x [1 − cos( k · x )] | g ( x ) | , (5.16)cf. [14, (5.32)]. We apply this bound with g ( x ) = Ψ z ( x ), combine it with (5.15) and (2.21), and useˆ C λ z ( l ± k ) ≥ / U λ z ( l, k ) in (2.16) to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ k ˆ b ( l )1 − ˆ a ( l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Kβ ˆ C λ z ( k ) − ˆ C λ z ( l ) ≤ O ( β ) U λ z ( l, k ) . (5.17)22he second term in (5.14) is bounded as follows. First, since | e il · x (e i ( ± k · x ) − | ≤ | sin( k · x ) | + 1 − cos( k · x ) , (5.18)we obtain (cid:12)(cid:12) ˆ b ( l ± k ) − ˆ b ( l ) (cid:12)(cid:12) = (cid:12)(cid:12) ˆΨ z ( l ± k ) − ˆΨ z ( l ) (cid:12)(cid:12) ≤ X x | sin( k · x ) | (cid:12)(cid:12) Ψ z ( x ) (cid:12)(cid:12) + X x [1 − cos( k · x )] (cid:12)(cid:12) Ψ z ( x ) (cid:12)(cid:12) . (5.19)The second term on the right hand side of (5.19) is bounded by O ( β ) ˆ C λ z ( k ) − ; on the first term weapply the Cauchy-Schwarz inequality and (2.20)–(2.21): X x | sin( k · x ) | (cid:12)(cid:12) Ψ z ( x ) (cid:12)(cid:12) ≤ (cid:16) X x =0 | Ψ z ( x ) | (cid:17) / (cid:16) X x =0 sin( k · x ) | Ψ z ( x ) | (cid:17) / ≤ O ( β ) / (cid:16) X x =0 [1 − cos( k · x )] | Ψ z ( x ) | (cid:17) / ≤ O ( β ) ˆ C λ z ( k ) − / . (5.20)Furthermore, ˆ a ( l ± k ) − ˆ a ( l ) = τ ( z ) (cid:16) ˆ D ( l ± k ) − ˆ D ( l ) (cid:17) + (cid:16) ˆΦ z ( l ± k ) − ˆΦ z ( l ) (cid:17) . (5.21)In a similar fashion as (5.19)–(5.20), we bound (cid:12)(cid:12)(cid:12) ˆΦ z ( l ± k ) − ˆΦ z ( l ) (cid:12)(cid:12)(cid:12) ≤ O ( β ) ˆ C λ z ( k ) − / and (cid:12)(cid:12)(cid:12) ˆ D ( l ± k ) − ˆ D ( l ) (cid:12)(cid:12)(cid:12) ≤ (cid:16) X x D ( x ) (cid:17) / (cid:16) X x [1 − cos( k · x )] D ( x ) (cid:17) / + X x [1 − cos( k · x )] D ( x )= 1 · [1 − ˆ D ( k )] / + [1 − ˆ D ( k )] ≤ C λ z ( k ) − / + 2 ˆ C λ z ( k ) − ≤ O (1) ˆ C λ z ( k ) − / , (5.22)where the last line uses (4.14). The combination of (5.19)–(5.22) and (5.7) yields (cid:0) ˆ a ( l ± k ) − ˆ a ( l ) (cid:1) (cid:0) ˆ b ( l ± k ) − ˆ b ( l ) (cid:1) ≤ O ( β ) ˆ C λ z ( k ) − . (5.23)On the other hand, by (5.12)–(5.13),11 − ˆ a ( l + σk ) = 1ˆ b ( l + σk ) ˆ G ( l + σk ) ≤ (1 + O ( β )) ˆ C λ z ( l + σk ) , σ ∈ {− , , } . (5.24)Combining (5.23) and (5.24) yields (cid:0) ˆ a ( l ± k ) − ˆ a ( l ) (cid:1) (cid:0) ˆ b ( l ± k ) − ˆ b ( l ) (cid:1) (1 − ˆ a ( l )) (1 − ˆ a ( l ± k )) ≤ O ( β ) ˆ C λ z ( k ) − ˆ C λ z ( l ) ˆ C λ z ( l ± k ) ≤ O ( β ) U λ z ( l, k ) . (5.25)For the third term in (5.14) we argue that | ˆ b ( l ) | = 1 + | ˆΨ z ( l ) | ≤ c β by our assumption on ˆΨ z .In order to apply Lemma 5.5 to bound ∆ k (1 − ˆ a ( l )) − , we estimateˆ A ( l ) := 11 − ˆ a ( l ) = 1ˆ b ( l ) ˆ G z ( l ) ≤ (1 + 2 c β ) (1 + 51 c β ) ˆ C λ z ( l ) ≤ (1 + Kβ ) ˆ C λ z ( l ) (5.26)23y Assumption 2.4 and (5.12), and c | a | (0) − c | a | ( k ) = X x [1 − cos( k · x )] (cid:12)(cid:12) τ ( z ) D ( x ) + Φ z ( x ) (cid:12)(cid:12) ≤ τ ( z )[1 − ˆ D ( k )] + X x [1 − cos( k · x )] (cid:12)(cid:12) Φ z ( x ) (cid:12)(cid:12) ≤ (2(1 + c β ) + c β ) ˆ C λ z ( k ) − ≤ C λ z ( k ) − , where the last line uses again (4.14) and, as usual, requires a certain smallness of β (here we need c β ≤ k − ˆ a ( l ) ≤ (1+ Kβ ) · · · ˆ C λ z ( k ) − n ˆ C λ z ( l − k ) ˆ C λ z ( l ) + ˆ C λ z ( l ) ˆ C λ z ( l + k ) + ˆ C λ z ( l − k ) ˆ C λ z ( l + k ) o , (5.27)so that finally | ∆ k ˆ G z ( l ) | U λ z ( k, l ) ≤ (1 + Kβ ) , (5.28)as required. In conclusion f ( z ) ≤ Kβ , and thus we obtain the improved bound f ( z ) ≤ O ( β ). Proof of Proposition 2.6.
Note first that f is continuous on (0 , z c ) by Lemma 5.3 and the assumedmean-field bound χ ( z ) ′ ≤ const χ ( z ) . Whence the prerequisites of Lemma 5.1 are satisfied by Lemma5.4 and the fact that f (0) = 1. Therefore, f ( z ) ≤ z < z c . Moreover, Lemma 5.4 shows that, if f ≤
4, then in fact f ≤ O ( β ). Hence f ( z ) ≤ O ( β ), uniformly for z < z c . 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 criticalexponent γ P for percolation exists, and satisfies γ P = 1. That is to say, they show χ ( z ) ≍ ( z c − z ) − as z ր z c . The lower bound γ P ≥ γ P ≤ T r := [ − r, r ] d ∩ Z d a cube of sidelength 2 r + 1. In order to achieve translation invariance, we equip the cube with periodicboundary conditions, that is, T r is a torus. In [8] free boundary conditions were used. We write G ( R ) z, T r ( 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 allbonds from T r with length larger than R . Define accordingly the restricted expected cluster size by χ ( R ) T r ( z ) := X x ∈ T r G ( R ) z, T r (0 , x ) , (A.1)and the restricted triangle diagram by ∇ ( R ) T r ( z ) := X v,s,t ∈ T r | v |≤ R D ( v ) G ( R ) z, T r ( v, s ) G ( R ) z, T r ( s, t ) G ( R ) z, T r ( t, . (A.2)24e proceed as follows. We fix ε > z < z c − ε , (cid:0) − ∇ ( R ) T r ( z c − ε ) − e R (cid:1) ( z c − z − ε ) ≤ χ ( R ) T r ( z ) − χ ( R ) T r ( z c − ε ) ≤ ( z c − z − ε ) (A.3)holds uniformly in r and R , where e R = o (1) as R → ∞ . We argue that indeed, for z < z c − ε ,lim R →∞ lim r →∞ χ ( R ) T r ( z ) = χ ( z ) , (A.4)and, for every R > ∇ ( R ) T r ( z c − ε ) ≤ ∇ ( z c − ε ) + o (1) as r ր ∞ , (A.5)where ∇ ( z ) = ( D ∗ G z ∗ G z ∗ G z )(0). Note that ∇ ( z ) differs from T ( z ) by the extra displacement D .Then, taking r → ∞ followed by R → ∞ , we obtain for every ε > − ∇ ( z c − ε )) ( z c − z − ε ) ≤ χ ( z ) − χ ( z c − ε ) ≤ z c − z − ε. (A.6)The limit ε ց − ∇ ( z c )) ( z c − z ) ≤ χ ( z ) ≤ z c − z. (A.7)since χ ( z c − ε ) − ց ε ց ∇ ( z c ) ≤ O ( β / ). Thus (A.7) implies γ P = 1 if β in Theorem 1.1 is sufficiently small, which sufficesfor our needs. It is possible to extend the argument to any finite triangle diagram (rather than smalltriangle diagrams only) by using ultraviolet regularization, as done in [8, Lemma 6.3].We start by proving (A.3). We call an (occupied or vacant) bond ( u, v ) pivotal for an increasingevent E , if E occurs if and only if ( u, v ) is occupied. A crucial tool in the proof is Russo’s formula [24,Theorem 2.25], stating thatdd z G ( R ) z, T r ( x, y ) = X ( u,v ) ∈ T r × T r | u − v |≤ R D ( v − u ) P ( R ) z, T r (( u, v ) is pivotal for x ↔ y ) , x, y ∈ T r . (A.8)The factor D ( v − u ) arises from the chain rule and the fact that the bond ( u, v ) is occupied withprobability zD ( v − u ). Since { ( u, v ) is pivotal for x ↔ y } ⊂ (cid:8) { x ↔ u } ◦ { v ↔ y } (cid:9) ∪ (cid:8) { x ↔ v } ◦ { u ↔ y } (cid:9) , (A.9)(A.8) and the BK-inequality [11] implydd z G ( R ) z, T r ( x, y ) ≤ X u,v ∈ T r D ( v − u ) P ( R ) z, T r ( x ↔ u ) P ( R ) z, T r ( v ↔ y ) . (A.10)Summing over y yields the upper bounddd z χ ( R ) T r ( z ) ≤ (cid:16) X u ∈ T r G ( R ) z, T r ( x, u ) (cid:17)(cid:16) X v ∈ T r D ( v − u ) (cid:17)(cid:16) X y ∈ T r G ( R ) z, T r ( v, y ) (cid:17) ≤ χ ( R ) T r ( z ) . (A.11)Therefore, dd z " − χ ( R ) T r ( z ) ≤ . (A.12)25ntegration over the interval ( z, z c − ε ) yields1 χ ( R ) T r ( z ) − χ ( R ) T r ( z c − ε ) ≤ z c − z − ε. (A.13)For the lower bound in (A.3) we use arguments as in [41, Section 9.4] to obtain P ( R ) z, T r (( u, v ) is pivotal for x ↔ y ) ≥ G ( R ) z, T r ( x, u ) G ( R ) z, T r ( v, y ) − X s,t ∈ T r G ( R ) z, T r ( x, t ) G ( R ) z, T r ( t, s ) G ( R ) z, T r ( t, u ) G ( R ) z, T r ( s, v ) G ( R ) z, T r ( s, y ) . (A.14)= − xt syvuyxu v (The contribution to the second line in (A.14) with u and v interchanged is hidden there, but isincorporated in the next line when we sum over both, u and v .) With Russo’s formula (A.8),dd z χ ( R ) T r ( z ) ≥ χ ( R ) T r ( z ) X | v |≤ R D ( v ) − χ ( R ) T r ( z ) X v,s,t ∈ T r | v |≤ R D ( v ) G ( R ) z, T r ( v, s ) G ( R ) z, T r ( s, t ) G ( R ) z, T r ( t, . (A.15)Since P v ∈ Z d D ( v ) = 1, the quantity e R := P | v | >R D ( v ) is o (1) as R → ∞ . Recalling the definition of ∇ ( R ) T r ( z ) in (A.2) we arrive atdd z " − χ ( R ) T r ( z ) ≥ (1 − e R ) − ∇ ( R ) T r ( z ) ≥ − ∇ ( R ) T r ( z c − ε ) − e R (A.16)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, T r |C| for the expected cluster size under themeasure P ( R ) z, T r , i.e., E ( R ) z, T r |C| = χ ( R ) T r ( z ). We further denote by ∂ R T r := T r + R \ T r the boundary of T r ofthickness R . Hence, E ( R ) z, T r + R |C| = E ( R ) z, T r + R |C| { = ∂ R T r } + E ( R ) z, T r + R |C| { ↔ ∂ R T r } . (A.17)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 leadsto E ( R ) z, T r + R |C| = E ( R ) z |C| − E ( R ) z |C| { ↔ ∂ R T r } + E ( R ) z, T r + R |C| { ↔ ∂ R T r } . (A.18)By the tree graph bound [8] and the monotonicity of E ( R ) z |C| in R , E ( R ) z |C| ≤ (cid:0) E ( R ) z |C| (cid:1) ≤ χ ( z ) , (A.19)and hence the Cauchy-Schwarz inequality yields E ( R ) z |C| { ↔ ∂ R T r } ≤ χ ( z ) / P z (0 ↔ ∂ R T r ) / . (A.20)For z < z c − ε , the first factor on the right is finite, and the latter vanishes as r → ∞ . For the lastsummand in (A.18), we bound as follows: E ( R ) z, T r + R |C| { ↔ ∂ R T r } ≤ (2( r + R ) + 1) d P ( R ) z, T r + R (0 ↔ ∂ R T r ) , (A.21)26ut, for r > R , P ( R ) z, T r + R (0 ↔ ∂ R T r ) ≤ P z, T r + R ( |C| ≥ r/R ) ≤ P z ( |C| ≥ r/R ) ≤ exp (cid:26) − r Rχ ( z ) (cid:27) , (A.22)where in the first bound we use the fact that occupied bonds have length ≤ R in the restricted model,the second bound utilizes the fact that clusters on the torus are a.s. smaller than clusters in the infinitelattice [30, Prop. 2.1], and in the third bound uses [8, Prop. 5.1]. The expression on the right hand sideof (A.22) decays exponentially as r increases, hence the right hand side of (A.21) vanishes and (A.4) isestablished once we have shown that E ( R ) z |C| → E z |C| as R → ∞ .This is done as follows. We write G ( R ) z and χ ( R ) for the model on the infinite lattice where bondsare restricted to have length ≤ R . Then obviously χ ( z ) ≥ χ ( R ) ( z ). Furthermore, G z ( x ) − G ( R ) z ( x ) = P z (0 ↔ x, ∃ pivotal bond ( u, v ) for { ↔ x } with | u − v | > R ) , hence, using the BK-inequality, χ ( z ) − χ ( R ) ( z ) ≤ χ ( z ) z X v : | v | >R D ( v ) . Again, this vanishes as R → ∞ , because z < z c − ε and P v D ( v ) = 1.It remains to prove (A.5). We use again the coupling of [30, Prop. 2.1] to write P ( R ) z c − ε, T r + R (0 ↔ x ) ≤ P z c − ε (0 ↔ x ) + P z c − ε (0 ↔ ∂ R T r ) . (A.23)Since the contribution from terms involving P z c − ε (0 ↔ ∂ R T r ) is again exponentially small in r (cf.(A.22)), we readily obtain (A.5). A.2 Derivation of δ P = 2 Barsky and Aizenman [10] showed that the triangle condition implies also β P = 1 and δ P = 2, wherethey used the general bounds β P ≤ δ P ≥ δ P is considered, namely ˆ δ P given by M ( z c , h ) := ∞ X k =1 [1 − e − kh ] P z c ( |C| = k ) ≍ h / ˆ δ P as h → ∞ . (A.24)The quantity M is known as magnetization . If we consider the critical exponents in terms of slowlyvarying functions only (and not our stronger version ≍ ), then the equivalence of δ P and ˆ δ P can be seendirectly via a Tauberian Theorem (e.g. [20, Theorem XIII.5.2]).Our version of δ P can be derived from (A.24), as we show now for the mean-field value δ P = 2. Inparticular, we show that c/ √ n ≤ M ( z c , /n ) ≤ C/ √ n, < c ≤ C < ∞ , (A.25)implies ˜ c/ √ n ≤ P z c ( |C| ≥ n ) ≤ ˜ C/ √ n for certain constants ˜ c, ˜ C ∈ (0 , ∞ ).For an upper bound on P z c ( |C| ≥ n ) we bound P z c ( |C| ≥ n ) = ∞ X k = n P z c ( |C| = k ) ≤ ∞ X k = n − e − k/n − e − P z c ( |C| = k ) ≤ (cid:2) − e − (cid:3) − ∞ X k =1 h − e − k/n i P z c ( |C| = k )= (cid:2) − e − (cid:3) − M ( p c , /n ) , (A.26)27nd hence P z c ( |C| ≥ n ) ≤ ˜ C/ √ n for ˜ C = [1 − e − ] − C .The lower bound is more involved. For every ε > P z c ( |C| ≥ n ) ≥ ∞ X k = n h − e − εk/n i P z c ( |C| = k )= M ( p c , ε/n ) − n − X k =1 h − e − εk/n i P z c ( |C| = k ) . We exploit 1 − e − x ≤ x to bound further n − X k =1 h − e − εk/n i P z c ( |C| = k ) ≤ εn n − X k =1 k P z c ( |C| = k ) . Note n − X k =1 k P z c ( |C| = k ) = n − X k =1 k X l =1 P z c ( |C| = k ) = n − X l =1 n − X k = l P z c ( |C| = k ) ≤ n − X l =1 P z c ( |C| ≥ l ) , whence P z c ( |C| ≥ n ) ≥ M ( p c , ε/n ) − εn n − X k =1 P z c ( |C| ≥ k ) . We apply (A.26) and compare with (A.25) to obtain P z c ( |C| ≥ n ) ≥ c √ ε √ n − εn n − X k =1 C [1 − e − ] √ k | {z } ≤ C [1 − e − ] − √ n . (A.27)This proves that P z c ( |C| ≥ n ) ≥ ˜ c/ √ n with ˜ c = c √ ε − εC [1 − e − ] − , and ˜ c > ε is smallenough. With a modification in (A.27), the argument can be extended to the case δ P = 2, but werefrain 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 consideringthe quantities π ( M ) Λ ( M = 0 , , , . . . ) defined in [38], which give rise to Π Λ M and R Λ M +1 by [38, (1.12) and(1.13)]: δ ,x + Π Λ M ( x ) = M X N =0 ( − N π ( N ) Λ ( x ) , ≤ (cid:12)(cid:12) R Λ M ( x ) (cid:12)(cid:12) ≤ τ ( z ) X u,v π ( M ) Λ ( u ) D ( v − u ) G ( v, x ) . (B.1)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 > . Then there exists a constant ¯ c K > , such that δ ,N ≤ X x π ( N ) Λ ( x ) ≤ ( c K β ( N = 0) , (¯ c K β ) N ( N ≥ , (B.2)28 nd X x [1 − cos( k · x )] π ( N ) Λ ( x ) ≤ ˆ C λ z ( k ) − (¯ c K β ) N ∨ , (B.3) uniformly in Λ . This proposition is a variation of [38, Proposition 3.2]. However, it is important that the bounds ofthe type P x | x | 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 explainswhy the factor | x | is not sufficient in the case of infinite variance spread-out models.By (B.27) below, π (0) z ( x ) ≤ G z ( x ) . Let us assume that G z ( x ) ≈ C λ z ( x ), as suggested by Theorem1.1. For z = z c , and using that C ( x ) ≈ const / | x | d − ( α ∧ , that would lead to X x | x | π (0) z c ( x ) ≈ X x | x | | x | d − ( α ∧ , and this is finite if and only if d < d − ( α ∧ −
2. In particular, this suggests that for α < α ∧ < d < / α ∧ P x | x | π (0) z c ( x ) = ∞ but P x [1 − cos( k · x )] π (0) z c ( x ) < ∞ . Thus, using P x | x | π (0) z c ( x ) < ∞ as a criterion for d > d c suggests a wrong value for the critical dimension. Rather,it appears that we must assume P x | x | α ∧ π (0) z c ( x ) < ∞ instead.We first show how Proposition B.1 implies Proposition 4.3, and afterwards discuss its proof. Proof of Proposition 4.3 subject to Proposition B.1.
We proceed as in the proof of [14, Prop. 5.2]. Thebounds (4.21)–(4.22) follow immediately with c K = 2¯ c K , where the extra 1 in the ( N = 0)-case iscompensated by the substraction of δ ,x , and the factor 2 comes from summing the geometric series(where we required β small enough to ensure ¯ c K β ≤ / R M ,we see by (B.1) that X x | R Λ M ( x ) | ≤ K ˆ π ( M ) Λ (0) χ ( z ) . (B.4)However, by (B.2), (4.23) follows if z < z c and M = M ( z ) is so large that ( c K β ) M χ ( z ) ≤ c K β . Finally,for (4.24), we use (B.57) below with j = 3 to see that X x ∈ Z d [1 − cos( k · x )] | R Λ M ( x ) | ≤ K [1 − ˆ D ( k )]ˆ π ( M ) Λ (0) χ ( z ) + 7 K (cid:0) ˆ π ( M ) Λ (0) − ˆ π ( M ) Λ ( k ) (cid:1) χ ( z )+ 7 K ˆ π ( M ) Λ (0) (cid:16) ˆ G z (0) − ˆ G z ( k ) (cid:17) . (B.5)For the first term, we use (4.14) and (B.2) to bound7 K [1 − ˆ D ( k )]ˆ π ( M ) Λ (0) χ ( z ) ≤ K (¯ c K β ) M χ ( z ) ˆ C λ z ( k ) − . For the second term, we use (B.3) to see that ˆ π ( M ) Λ (0) − ˆ π ( M ) Λ ( k ) ≤ ˆ C λ z ( k ) − (¯ c K β ) M ∨ . Finally, for thethird term in (B.5), we use the upper bound on f and the uniform bound ˆ C λ z ( k ) ≤ (1 − λ z ) − = χ ( z )to obtain | ˆ G z (0) − ˆ G z ( k ) | = 12 | ∆ k ˆ G z (0) | ≤ K ˆ C λ z ( k ) − (cid:16) − λ z ) − (cid:17) = 48 K ˆ C λ z ( k ) − χ ( z ) . (B.6)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 theremainder of the paper. Our proof is an adaptation of the proof of [38, Prop. 3.2], with a modifiedbootstrap hypothesis. In particular, the factor | x | at various places in that proof is replaced by the29actor 1 − cos( k · x ) here. We fix z ∈ (0 , z c ) and throughout the remainder of the section omit it from thenotation (e.g., we write τ for τ ( z )). Also we fix some subset Λ containing the origin. We keep in mindthat 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 define the quantity ˜ G ( x ) := τ ( D ∗ G )( x ) , (B.7)and note the basic estimate G ( x ) ≤ δ ,x + ˜ G ( x ) (B.8)resulting from the random-current representation and the source switching lemma (cf. [38, (4.2)]).In line with (1.38), we write B = ( G ∗ G )(0) = P x G ( x ) for the bubble diagram , and similarly˜ B = ( ˜ G ∗ ˜ G )(0) for the “non-vanishing bubble diagram”. For the latter we bound˜ B = τ Z [ − π,π ) d (cid:16) ˆ D ( k ) ˆ G ( k ) (cid:17) d k (2 π ) d ≤ K Z [ − π,π ) d (cid:16) ˆ D ( k ) ˆ C λ z ( k ) (cid:17) d k (2 π ) d ≤ K Z [ − π,π ) d ˆ D ( k ) [1 − ˆ D ( k )] d k (2 π ) d ≤ K β using that τ = f ( z ) ≤ K and f ( z ) ≤ K in the first line, and (4.14) and Assumption 2.1 in the secondline. On the other hand, by (B.8), B = X x G ( x ) = 1 + X x =0 G ( x ) ≤ X x ˜ G ( x ) = 1 + ˜ B ≤ K β. (B.9)Furthermore, it is easy to see that, by the Cauchy-Schwarz inequality, “open bubbles” are bounded bya “closed bubble”, i.e., for all x ∈ Z d ,( G ∗ G )( x ) = X v G ( v ) G ( x − v ) ≤ B, ( ˜ G ∗ ˜ G )( x ) ≤ ˜ B. (B.10)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 sufficientlysmall.We now define various quantities needed to describe the bounding diagrams. All notation is chosenconsistently with [38], which provides our basic estimates. In order to emphasize the diagrammaticstructure, we write G and ˜ G with two arguments, with the understanding that G ( y, x ) = G ( x − y ), andfor ˜ G appropriately.Let ψ ( y, x ) := ∞ X j =0 ( ˜ G ) ∗ j ( y, x ) = δ y,x + ∞ X j =1 X u ,u ,...,u j ∈{ x }× ( Z d ) j − ×{ y } j Y l =1 ˜ G ( u l − , u l ) (B.11)denote a “chain of bubbles”, and ˜ ψ ( y, x ) = ψ ( y, x ) − δ y,x . (B.12)If β is so small that ˜ B < / ψ := sup y X x ˜ ψ ( y, x ) ≤ B = O ( β ) . (B.13)30et P ′ (0) u ( y, x ) := G ( y, x ) G ( y, u ) G ( u, x ) = y u x , (B.14) P ′′ (0) u,v ( y, x ) := G ( y, x ) G ( y, u ) G ( u, x ) X v ′ G ( y, v ′ ) G ( v ′ , x ) ψ ( v ′ , v ) = y u x(v ) ’ v . (B.15)In the last equalities of (B.14)–(B.15) we used the pictorial representation introduced in Figure 1. Recallthat a line between two points, say y and x , represents the two-point function G ( y, x ), and vertices inbrackets are summed over. The quantities P ′ (0) and P ′′ (0) are the leading terms in the quantities P ′ and P ′′ , defined in (B.22) below.We further define P (1) ( v , v ′ ) := 2 ˜ ψ ( v , v ′ ) G ( v , v ′ ) , (B.16)and, for j = 2 , , . . . , P ( j ) ( v , v ′ j ) := X v ,...,v j v ′ ,...,v ′ j − G ( v , v ) G ( v , v ′ ) j Y i =1 ˜ ψ ( v , v ′ ) ! × j − Y i =2 G ( v ′ i − , v i +1 ) G ( v i +1 , v ′ i ) ! G ( v j , v ′ j − ) . (B.17)The first three elements of the sequence look diagrammatically like P (1) ( v , v ′ ) = v v ’ , P (2) ( v , v ′ ) = (v ’ ) v ’ (v ) v , P (3) ( v , v ′ ) = v (v ) (v ’ ) (v ’ ) (v ) v .Recall that vertices in brackets are summed over.We now obtain quantities P ′ and P ′′ as variations on P . To this end, we define P ′ ( j ) u ( v , v ′ j ) byreplacing one of the 2 j − G ( z, z ′ ), on the right-hand side of (B.16)–(B.17)by the product of two two-point functions, G ( z, u ) G ( u, z ′ ), and then summing over all 2 j − P ′ (1) u ( v , v ′ ) = 2 ˜ ψ ( v , v ′ ) G ( v , u ) G ( u, v ′ ) = v u v ’ , (B.18)and P ′ (2) u ( v , v ′ ) = X v ,v ′ (cid:18) Y i =1 ˜ ψ ( v i , v ′ i ) (cid:19)(cid:16) G ( v , u ) G ( u, v ) G ( v , v ′ ) G ( v ′ , v ′ )+ G ( v , v ) G ( v , u ) G ( u, v ′ ) G ( v ′ , v ′ )+ G ( v , v ) G ( v , v ′ ) G ( v ′ , u ) G ( u, v ′ ) (cid:17) . (B.19)We define P ′′ ( j ) u,v ( v , v ′ j ) similarly as follows. First we take two two-point functions in P ( j ) ( v , v ′ j ), oneof which (say, G ( y , y ′ ) for some y , y ′ ) is among the aforementioned 2 j − G ( y , y ′ ) for some y , y ′ ) is among those of which ψ ( v i , v ′ i ) − δ v i ,v ′ i for i = 1 , . . . , j arecomposed. The product G ( y , y ′ ) ˜ G ( y , y ′ ) is then replaced by (cid:18) X v ′ G ( y , v ′ ) G ( v ′ , y ′ ) ψ ( v ′ , v ) (cid:19)(cid:16) G ( y , u ) ˜ G ( u, y ′ ) + ˜ G ( y , y ′ ) δ u,y ′ (cid:17) + G ( y , u ) G ( u, y ′ ) X v ′ (cid:16) G ( y , v ′ ) ˜ G ( v ′ , y ′ ) + ˜ G ( y , y ′ ) δ v ′ ,y ′ (cid:17) ψ ( v ′ , v ) . (B.20)In our pictorial representation, y y ’ y y ’ is replaced by v(v ) ’ u y y ’ y y ’ + v(v ) ’ u y y ’ y y ’ .Finally, we define P ′′ ( j ) u,v ( v , v ′ j ) by taking account of all possible combinations of G ( y , y ′ ) and ˜ G ( y , y ′ ).For example, we define P ′′ (1) u,v ( v , v ′ ) as P ′′ (1) u,v ( v , v ′ ) = X u ′ ,u ′′ ,v ′ (cid:18) ψ ( v , u ′ ) ˜ G ( u ′ , u ′′ ) (cid:16) G ( u ′ , u ) ˜ G ( u, u ′′ ) + ˜ G ( u ′ , u ′′ ) δ u,u ′′ (cid:17) ψ ( u ′′ , v ′ ) × G ( v , v ′ ) G ( v ′ , v ′ ) ψ ( v ′ , v ) + (permutation of u and v ′ ) (cid:19) (B.21)= v u v ’ (v ) ’ v + v u v ’ (v ) ’ v ,where the permutation term corresponds to the second diagram.We let P ′ u ( y, x ) = X j ≥ P ′ ( j ) u ( y, x ) = yu x , P ′′ u,v ( y, x ) = X j ≥ P ′′ ( j ) u,v ( y, x ) = yu xv , (B.22)where P ′ (0) u ( y, x ) and P ′′ (0) u,v ( y, x ) are the leading contributions to P ′ u ( y, x ) and P ′′ u,v ( y, x ), respectively.Finally, we define Q ′ u ( y, x ) = X z (cid:0) δ y,z + ˜ G ( y, z ) (cid:1) P ′ u ( z, x ) = yu x(z) , (B.23) Q ′′ u,v ( y, x ) = X z (cid:0) δ y,z + ˜ G ( y, z ) (cid:1) P ′′ u,v ( z, x )+ X v ′ ,z (cid:0) δ y,v ′ + ˜ G ( y, v ′ ) (cid:1) ˜ G ( v ′ , z ) P ′ u ( z, x ) ψ ( v ′ , v ) , (B.24)that is, pictorially, Q ′′ u,v ( y, x ) = y u x v Q ’’ = yu xv(z) + yu xv(v ’ (z) ) . (B.25)Based on the lace expansion, Sakai proved the following diagrammatic bound:32 roposition B.2 (Diagrammatic bounds [38, Prop. 4.1]) . For the ferromagnetic Ising model, π ( N ) Λ ( x ) ≤ P ′ (0) (0 , x ) ( N = 0) , X b ,...,b j v ,...,v j P ′ (0) v (0 , b ) N − Y i =1 τ D ( b i ) Q ′′ v i ,v i +1 ( b i , b i +1 ) ! τ D ( b j ) Q ′ v i ,v i +1 ( b i , x ) ( N ≥ , (B.26) where the sum is taken over vertices v i and (directed) bonds b i = ( b i , b i ) , i = 1 , . . . , j . We denote D ( b i ) = D ( b i − b i ) and regard the empty product as 1 by convention. The bound (B.26) holds uniformlyin Λ . It should be noted that Sakai [38] proved the bound (B.26) on a finite graph Λ, where in particularall quantities on the right hand side are defined on Λ. By Griffith’s second inequality [23], the two-pointcorrelation function G z is monotonically increasing in Λ, and thus so are P ′ , Q ′ and Q ′′ . Hence, the righthand 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 thermodynamiclimit 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 ≤ P x π (0) Λ ( x ) ≤ O ( β ). By the definition of π (0) Λ ( x ) and (B.14), δ ,x ≤ π (0) Λ ( x ) ≤ G ( x ) . Whence1 ≤ X x π (0) Λ ( x ) ≤ X x =0 G ( x ) ≤ sup x =0 G ( x ) ! X x =0 ˜ G ( x ) . (B.27)The term P x =0 ˜ G ( 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 obtainsup x =0 G ( x ) ≤ τ k D k ∞ + k τ D ∗ ˜ G k ∞ . (B.28)The first summand is bounded by Kβ , by our bound on f and (2.9). Furthermore, k τ D ∗ ˜ G k ∞ ≤ K β by a calculation similar to (B.9) and using 1 ≤ − ˆ D ( k )] − . We thus obtain the bound on P x π (0) Λ ( x ).We next consider the bound on P x π ( N ) Λ ( x ) for N ≥
1. Here is a diagrammatic representation ofthe bounds on P x π ( N ) Λ ( x ) for N = 3:where all vertices v , v , v and bonds b , b , b are summed over. Since the diagrammatic bound(B.26) implies X x π ( N ) Λ ( x ) ≤ X v,x P ′ (0) v (0 , x ) ! sup y X w,v,x τ D ( w − y ) Q ′′ ,v ( w, x ) ! N − sup y X w,x τ D ( w − y ) Q ′ ( w, x ) ! , (B.29)it is sufficient to show that(i) P v,x P ′ (0) v (0 , x ) ≤ O (1), 33ii) sup y P w,x τ D ( w − y ) Q ′ ( w, x ) ≤ O ( β ),(iii) sup y P w,v,x τ D ( w − y ) Q ′′ ,v ( w, x ) ≤ O ( β ).We will now prove these bounds one at a time.(i) We first show that P v,x P ′ (0) v (0 , x ) is uniformly bounded. Indeed, by (B.10) and (B.14), X v,x P ′ (0) v (0 , x ) = X v,x G ( x ) G ( v ) G ( v − x ) ≤ sup y X v G ( v ) G ( v − y ) ! X x G ( x ) ≤ B . (B.30)(ii) We bound X w,x τ D ( w − y ) Q ′ ( w, x ) = X u,x X w τ D ( w − y ) (cid:0) δ w,u + ˜ G ( u − w ) (cid:1)! P ′ ( u, x ) , (B.31)cf. (B.23). The factor β comes from the nonzero line segment P w τ D ( w − y ) (cid:0) δ w,u + ˜ G ( u − w ) (cid:1) , as wehave seen in the discussion around (B.28).It remains to show that P u,x P ′ ( u, x ) = P u,x P ∞ j =0 P ′ ( j ) ( u, x ) is uniformly bounded. Claim B.3 (Bound on P ′ ) . X u,x P ′ ( u, x ) ≤ O (1) . (B.32) Proof.
To this end, it suffices to show X u,x P ′ ( j ) ( u, x ) ≤ (2 j − O ( β ) j , ( j ≥ , (B.33)since the case j = 0 has been treated in (B.30). The bound (B.33) will be achieved by decomposing thediagrams describing P ′ ( j ) into bubble diagrams, and we demonstrate this for the case j = 4 explicitly.Recall from (B.17) that P (4) ( u, x ) = u x , (B.34)and we obtain P ′ (4) ( u, x ) from P (4) ( u, x ) by replacing one of the 7(= 2 j −
1) factors of the form G ( u, v )by P w G ( u, w ) G ( w, v ). In terms of diagrams, there is an extra vertex added to either of the 7 straightlines in (B.34). This explains the factor (2 j −
1) in (B.33).In case this extra vertex falls to one of the horizontal lines, say the lower one, we bound as follows.We first extend our diagrammatical notation in the following way: we mark vertices that are summedover by a full dot, and fixed vertices (possibly with a supremum) are marked with an open dot, i.e., X u,x u x0 = .34y multiple use of translation invariance of the model, we obtain= X x ,x ,x ,x ,x ,x ,x ,x x x x x x x x x y = X x ,x ,x ,x ,x ,x ,x ,x G ( x , x ) G ( x , x ) G ( x , y ) G ( y, x ) G ( x , x ) G ( x , x ) × G ( x , x ) G ( x , x ) ˜ ψ ( x , x ) ˜ ψ ( x , x ) ˜ ψ ( x , x ) ˜ ψ ( x , x )= X x ,x ,y,x ,x ,x ,x ,x · · · (expression as above with x fixed) ≤ X x ˜ ψ ( x , x ) ! sup ¯ x X x G (¯ x , x ) G ( x , x ) ! × sup ¯ x X x ˜ ψ (¯ x , x ) ! sup ¯ x X y G ( x , y ) G ( y, ¯ x ) ! × sup x X x ,x ,x ,x G ( x , x ) G ( x , x ) G ( x , x ) G ( x , x ) ˜ ψ ( x , x ) ˜ ψ ( x , x ) ! = (B.35)For the remaining component on the right hand side, we again use translation invariance and boundfurther as = ≤ , = ≤ ≤ ˜ ψ B. (B.36)Hence, ≤ B ˜ ψ , (B.37)and this can be made smaller than O ( β ) , cf. (B.9) and (B.13).However, if the extra vertex falls to one of the vertical lines, then the details are slightly different:= ≤ ≤ B ˜ ψ . (B.38)The remaining diagram in (B.38) is bounded by multiple use of translation invariance, as we will show35ow: = sup w X v,x,y,z = sup w X x,y,z,v ≤ sup w,y X v ! X x,y,z ≤ B · ˜ ψ . (B.39)This proves (B.33) for j = 4. The cases j , } are omitted, since the same methods will lead tothe desired bounds.(iii) We now turn to the bounds involving Q ′′ , i.e., we provesup y X w,v,x τ D ( w − y ) Q ′′ ,v ( w, x ) ≤ O ( β ) . (B.40)Recalling the definition of Q ′′ in (B.24), (B.40) is established once we have shownsup y X w,v,v ′ ,z,x τ D ( w − y ) (cid:0) δ w,v ′ + ˜ G ( w, v ′ ) (cid:1) ˜ G ( v ′ , z ) P ′ ( z, x ) ψ ( v ′ , v ) ≤ O ( β ) (B.41)and sup y X w,v,z,x τ D ( w − y ) (cid:0) δ w,z + ˜ G ( w, z ) (cid:1) P ′′ ,v ( z, x ) ≤ O ( β ) . (B.42)A decomposition of the left hand side of (B.41) yields as an upper bound sup z X w,v ′ τ D ( w − y ) (cid:0) δ w,v ′ + ˜ G ( w, v ′ ) (cid:1) ˜ G ( v ′ , z ) sup v ′ X v ψ ( v ′ , v ) ! X z,x P ′ ( z, x ) ! , (B.43)where the first term is bounded by O ( β ), the second term is bounded by 1 + ˜ ψ = O (1) and the finalterm 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 showsup y X w,v,z,x τ D ( w − y ) (cid:0) δ w,z + ˜ G ( w, z ) (cid:1) P ′′ ( j ) ,v ( z, x ) ≤ O ( β ) j ∨ (B.45)for j = 0 , , , . . . . We explicitly perform this bound for j = 0 ,
1, and omit the details for j ≥ j = 0, we bound ≤ , (B.46)i.e., sup y X w,v,z,x τ D ( w − y ) (cid:0) δ w,z + ˜ G ( w, z ) (cid:1) P ′′ (0) ,v ( z, x ) ≤ O ( β ) B (1 + ˜ ψ ) , (B.47)where the O ( β )-factor arises from the open bubble involving the extra vertex, and the chain of bubbleshanging off from the top produces a factor 1 + ˜ ψ .For j = 1 we proceed similarly by recalling the definition of P ′′ (1) in (B.21) and boundsup y X w,v,z,x τ D ( w − y ) (cid:0) δ w,z + ˜ G ( w, z ) (cid:1) P ′′ (1) ,v ( z, x ) = + ≤ + ,where the numbers indicate the order in the decomposition. A calculation similar to (B.39) shows that ≤ ( ) ( ) = B (1 + ˜ ψ ) (if the initial two-point function is dashed, then we obtain˜ B (1 + ˜ ψ ) as an upper bound). Hence (B.45) for j = 1 follows. The terms for j ≥ Proof of (B.3).
We now turn towards the proof of the bound (B.3) in Proposition B.1, which we restatehere for convenience: X x [1 − cos( k · x )] π ( N ) Λ ( x ) ≤ ˆ C λ z ( k ) − (¯ c K β ) N ∨ . We start by considering the case N = 0. By (B.26) and (B.14), X x [1 − cos( k · x )] π (0) Λ ( x ) ≤ X x =0 [1 − cos( k · x )] G ( x ) . (B.48)This is bounded above by (cid:18) sup x [1 − cos( k · x )] G ( x ) (cid:19) X x =0 G ( x ) ≤ (cid:18) sup x [1 − cos( k · x )] G ( x ) (cid:19) ˜ B. (B.49)Then the desired bound follows from (B.9) and the following lemma: Lemma B.5.
If for some model we have that f ( z ) ≤ K for some z ∈ (0 , z c ) , K > , then sup x [1 − cos( k · x )] G ( x ) ≤ K ˆ C λ z ( k ) − ( C λ z ∗ C λ z )(0) . (B.50)Casually speaking, the multiplication by [1 − cos( k · x )] yields a factor ˆ C λ z ( k ) − at the expense ofadding an extra vertex in the bounding ( C -)diagram. In fact, we need only that ˆ C λ z ( k ) − O (1) is anupper bound. Although the lemma is applied to the Ising model here, it is valid for any model as longas f ( z ) ≤ K . 37 roof of Lemma B.5. Sincesup x [1 − cos( k · x )] G ( x ) = sup x Z [ − π,π ) d e − il · x (cid:18) ˆ G z ( l ) − (cid:16) ˆ G z ( l − k ) + ˆ G z ( l + k ) (cid:17)(cid:19) d l (2 π ) d ≤ Z [ − π,π ) d (cid:12)(cid:12)(cid:12)(cid:12) ˆ G z ( l ) − (cid:16) ˆ G z ( l − k ) + ˆ G z ( l + k ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) d l (2 π ) d = Z [ − π,π ) d (cid:12)(cid:12)(cid:12)(cid:12)
12 ∆ k ˆ G ( l ) (cid:12)(cid:12)(cid:12)(cid:12) d l (2 π ) d , (B.51)our bound f ≤ K implies thatsup x [1 − cos( k · x )] G ( x ) ≤ K ˆ C λ z ( k ) − Z [ − π,π ) d (cid:16) ˆ C λ z ( l − k ) ˆ C λ z ( l + k ) + ˆ C λ z ( l − k ) ˆ C λ z ( l ) + ˆ C λ z ( l ) ˆ C λ z ( l + k ) (cid:17) d l (2 π ) d . (B.52)Denoting C λ z ,k ( x ) := cos( k · x ) C λ z ( x ), we observe that | C λ z ,k ( x ) | ≤ C λ z ( x ) andˆ C λ z ,k ( l ) = 12 (cid:16) ˆ C λ z ( l − k ) + ˆ C λ z ( l + k ) (cid:17) . (B.53)Hence, Z [ − π,π ) d (cid:16) ˆ C λ z ( l − k ) ˆ C λ z ( l ) + ˆ C λ z ( l ) ˆ C λ z ( l + k ) (cid:17) d l (2 π ) d = 2 Z [ − π,π ) d ˆ C λ z ( l ) ˆ C λ z ,k ( l ) d l (2 π ) d (B.54)= 2( C λ z ∗ C λ z ,k )(0) ≤ C λ z ∗ C λ z )(0) . Furthermore,ˆ C λ z ( l − k ) ˆ C λ z ( l + k ) = 14 h ˆ C λ z ( l − k ) + ˆ C λ z ( l + k ) i − h ˆ C λ z ( l − k ) − ˆ C λ z ( l + k ) i ≤ h ˆ C λ z ( l − k ) + ˆ C λ z ( l + k ) i = ˆ C λ z ,k ( l ) , (B.55)so that Z [ − π,π ) d ˆ C λ z ( l − k ) ˆ C λ z ( l + k ) d l (2 π ) d ≤ Z [ − π,π ) d ˆ C λ z ,k ( l ) d l (2 π ) d = ( C λ z ,k ∗ C λ z ,k )(0) ≤ ( C λ z ∗ C λ z )(0) . (B.56)The combination of the above inequalities implies the claim.For N >
0, our strategy is to break the term 1 − cos( k · x ) into parts using1 − cos t ≤ (2 N + 3) N X n =0 [1 − cos t n ] for t = N X n =0 t n (B.57)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 boundon π (1) Λ ( x ). An application of (B.57) for N = 1 yields X x [1 − cos( k · x )] π (1) Λ ( x ) ≤ X x [1 − cos( k · x )] ≤ + ! . (B.58)38n (B.58) we extend our pictorial representation to incorporate factors of the form [1 − cos( k · x )]. Herea double line between two points, say y and y , represents a factor [1 − cos( k · ( y − y ))] G ( y − y ),while, as before, a normal line represents a factor G ( y − y ). For the second summand in (B.58) , thereis not a single two-point function between the two endpoints of the double line. Here our understandingis that = X x,y [1 − cos( k · ( x − y ))] . (B.59)In other words, the double line between the two points y and x gives rise to the factor [1 − cos( k · ( x − y ))].The first term in (B.58) is estimated like ≤ (i) (iii) (iv)(ii) , (B.60)which yields factors B ˆ C λ z ( k ) − arising from (i) by Lemma B.5, B from (ii) , ˜ B from (iii) , and O (1)from (iv) by Claim B.3. Thus, ≤ ˆ C λ z ( k ) − O ( β ) . (B.61)For the second term in (B.58) we bound ≤ ≤ B · , (B.62)The remaining factor sup y P w,x [1 − cos( k · x )] τ D ( w − y ) Q ′ ( w, x ) is bounded by the following claim: Claim B.6.
Under the assumptions of Proposition B.1, = sup y X w,x [1 − cos( k · x )] τ D ( w − y ) Q ′ ( w, x ) ≤ ˆ C λ z ( k ) − O ( β ) . (B.63) Proof.
By (B.23),sup y X w,x [1 − cos( k · x )] τ D ( w − y ) Q ′ ( w, x ) = sup y X w,x,z [1 − cos( k · x )] τ D ( w − y ) ∞ X j =0 (cid:16) δ w,z + ˜ G z ( w, z ) (cid:17) P ′ ( j ) ( z, x ) . (B.64)In diagrams, that is ≤ + + + + ! + · · · , (B.65)where contributions according to j = 0 , , − cos( k · ( y − y ))], where y is the starting point of the lines, and y is theendpoint. We then use (B.57) to decompose the series of double lines. For example, for the first termin parenthesis we obtain ≤ + + ! , j = 2 in (B.65) (the term in parenthesis) is bounded by O ( β ) ˆ C λ z ( k ) − . The method can be generalizedto j ≥ y X w,x,z [1 − cos( k · x )] τ D ( w − y ) (cid:16) δ w,z + ˜ G z ( w, z ) (cid:17) P ′ ( j ) ( z, x ) ≤ O ( j ) O ( β ) j +1 ˆ C λ z ( k ) − . (B.66)By (B.64), this is sufficient for (B.63).For N >
1, we proceed by distributing the spatial displacement 1 − cos( k · x ) along the “bottomline” of the diagram. E.g., for N = 3, this yields X x [1 − cos( k · x )] π (3) Λ ( x ) ≤ X x [1 − cos( k · x )] Q ’’ Q ’’ = . (B.67)By (B.57), the right hand side of (B.67) is bounded above by 9 times++ + . (B.68)In the following we refer by (I), (II), (III) and (IV) to the four terms in (B.68), respectively. In fact,all 4 terms are bounded by ˆ C λ z ( k ) − O ( β ) N , as we will show now.The bound on (IV) is an immediate consequence of (i) and (iii) below (B.29), and Claim B.6. Forthe bound on (I), we use translation invariance to obtain the factorization | {z } (I-1) Q ’’ | {z } (I-2) Q ’’ | {z } (I-3) | {z } (I-4) . (B.69)The terms indicated by ←− Q ′′ in the diagram are obtaines from Q ′′ by shifting the two-point functionshanging off the left side of the Q ′′ -box to the next factor on the left hand side, i.e. (compare with (B.25)) ←− Q ′′ ,v ( y, x ) = X z,z ′ P ′′ ,v ( y, z ) τ D ( z ′ − z ) (cid:0) δ z ′ ,x + ˜ G ( z ′ , x ) (cid:1) + X z,z ′ ,w ˜ G ( y, w ) P ′ ( w, z ) ψ ( y, v ) τ D ( z ′ − z ) (cid:0) δ z ′ ,x + ˜ G ( z ′ , x ) (cid:1) (B.70)= Q ’’ vy x0 z z ’ .The first factor (I-1) is bounded by ˆ C λ z ( k ) − O ( β ) as in (B.60). The middle terms (I-2) and (I-3)are equal to sup x P v,y ←− Q ′′ ,v ( y, x ). Performing calculations as in (B.40)–(B.42), it can be shown thatactually sup x X v,y ←− Q ′′ ,v ( y, x ) ≤ O ( β ) , (B.71)40nd this term occurs N − O (1), cf. Claim B.3. Thebounds on (I-1)–(I-4) show that (I) ≤ ˆ C λ z ( k ) − O ( β ) N .The terms (II) and (III) are bounded in a similar fashion by product structures:(II) ≤ Q ’’ Q ’’ Q ’’ , (B.72)(III) ≤ Q ’’ Q ’’ . (B.73)The term P v,x P ′ (0) v (0 , x ) on the left hand side is bounded by O (1) by (B.30); the term P u,x P ′ ( u, x )(the gray triangle on the right) is bounded by O (1) by Claim B.3. The terms involving Q ′′ and ←− Q ′′ arebounded by O ( β ) by (B.40) and (B.71), and together there are N − Q ’’ ≤ ˆ C λ z ( k ) − O ( β ) , Q ’’ ≤ ˆ C λ z ( k ) − O ( β ) . (B.74)Here the dashed arrow indicates that the supremum is taken over the difference between the twovertices 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. First we use (B.57) to distribute the spatial displacement of 1 − cos( k · x ) tosingle two-point functions G or ˜ G . Secondly, from each of the emerging summands, we eliminate theterm of the form sup x,y [1 − cos( k · ( y − x ))] G ( x, y ) (where x and y are chosen appropriately), and boundit by ˆ C λ z ( k ) − O (1), cf. Lemma B.5. Finally, we bound the remaining quantity in the same fashionas in (B.40)–(B.42). Note that the removed bond is compensated by an extra bond hanging off thelower / upper right corner. The factor β arises from the bubbles involving the two non-zero two-pointfunctions hanging off the box. This finally leads to the required bound(II) + (III) ≤ ˆ C λ z ( k ) − O ( β ) N , (B.75)and thus proves (B.3). This completes the proof of Proposition B.1. Acknowledgement.
We thank Roberto Fern´andez and Aernout van Enter for inspiring discussions.This work was supported by the Netherlands Organization for Scientific Research (NWO). MH visitedthe University of Bath on a grant from the RDSES programme of the European Science Foundation.
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