Perturbative analysis of disordered Ising models close to criticality

Transcription

1 Perturative analysis of disordered Ising models close to criticality Lorenzo Bertini 1 Emilio N.M. Cirillo 2 Enzo Olivieri 3 1 Dipartimento di Matematica, Università di Roma La Sapienza Piazzale Aldo Moro 2, Roma, Italy E mail: ertini@mat.uniroma1.it 2 Dipartimento Me. Mo. Mat., Università di Roma La Sapienza Via A. Scarpa 16, Roma, Italy E mail: cirillo@dmmm.uniroma1.it 3 Dipartimento di Matematica, Università di Roma Tor Vergata Via della Ricerca Scientifica, Roma, Italy E mail: olivieri@mat.uniroma2.it Astract We consider a two dimensional Ising model with random i.i.d. nearest neighor ferromagnetic couplings and no external magnetic field. We show that, if the proaility of supercritical couplings is small enough, the system admits a convergent cluster expansion with proaility one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the aove expansion implies the infinite differentiaility of the free energy ut not its analyticity. The asic tool in the proof are a general theory of graded cluster expansion and a stochastic domination of the disorder. MSC2000. Primary 82B44, 60K35. Keywords and phrases. Ising models, Disordered systems, Cluster expansion, Griffiths singularity. The authors acknowledge the partial support of Cofinanziamento PRIN.

2 1. Introduction and main result In [2] we developed a general theory concerning a graded perturative expansion for a class of lattice spin systems. This theory is useful when the system deserves a multi scale description namely, when a recursive analysis is needed on increasing length scales. The typical example is provided y a disordered system, like a quenched spin glass, having a good ehavior in average ut with the possiility of aritrarily large ad regions. Here y good ehavior we mean the one of a weakly coupled random field. In the ad (i.e. not good) regions the system can e instead strongly correlated. If those ad regions are suitaly sparse then the good ones ecome dominant allowing an analysis ased on an iterative procedure. Consider Ising systems descried y the following formal Hamiltonian which includes the inverse temperature H(σ) = J x,y σ x σ y h σ x (1.1) x x,y Z 2 x y =1 where σ x { 1, +1}, h R, and J x,y are i.i.d. random variales. A well known example is the Edwards Anderson model [6] defined y choosing J x,y centered Gaussian random variales with variance s 2. Let h = 0; if s 2 is small enough, so that the proaility of sucritical couplings is close to one, we expect a weak coupling regime. However, in the infinite lattice Z 2 there are, with proaility one, aritrarily large regions where the random couplings take large positive values giving rise, inside these regions, to the ehavior of a low temperature ferromagnetic Ising system with long range order. A simpler system is the so called diluted Ising model, defined y choosing J x,y equal to K > 0 with proaility q and to zero with proaility 1 q. In this case it is possile to show that for q sufficiently small and K sufficiently large the infinite volume free energy is infinitely differentiale ut not analytical in h [9,18]. This is a sort of infinite order phase transition called Griffiths singularity. A similar ehavior is conjectured for a general distriution of the random couplings whenever the proaility of supercritical values is sufficiently small ut strictly positive. More precisely, in such a situation, we expect an exponential decay of correlations with a non random decay rate ut with a random unounded prefactor. This should induce infinite differentiaility of the limiting quenched free energy, ut the presence of aritrarily large regions with supercritical couplings should cause the reaking of analyticity. The rigorous analysis of disordered systems in the Griffiths phase goes ack to [7], where a powerful and widely applicale perturative expansion has een introduced. The typical applications are high temperature spin glasses and random field Ising models with large variance. From now on we focus however on ferromagnetic random Ising systems with ounded couplings J x,y and h = 0. Let K c e the critical coupling for the standard two dimensional Ising model and K 1 < K c e such that for coupling constants J x,y [0, K 1 ] the standard high temperature cluster expansion is convergent, see e.g (i) in [8]. In the context of [7] a ond {x, y} is to e considered ad if the corresponding coupling J x,y exceeds the value K 1. The theory developed in [7] is ased on a multi scale 1

3 classification of the ad onds yielding that, with high proaility, larger and larger ad regions are farther and farther apart. In particular there exists a constant q 1 (0, 1) such that if Pro(J x,y > K 1 ) q 1 then the system admits a convergent graded cluster expansion implying the exponential decay of correlations as stated aove. The aim of the present paper is to analyze disordered systems that are weakly coupled only on a sufficiently large scale depending on the thermodynamic parameters. In particular we consider random ferromagnetic Ising models allowing typical values of the coupling constants aritrarily close to the critical value K c. In this case we need a graded cluster expansion ased on a scale adapted approach. To introduce this notion let us take for a while the case of the deterministic ferromagnetic Ising model with coupling K smaller than the critical value K c. The standard high temperature expansions, converging for coupling smaller than K 1, involve perturations around a universal reference system consisting of independent spins. In [14, 15] another perturative expansion has een introduced, around a not trivial model dependent reference system, that can e called scale adapted. Its use is necessary if we want to treat perturatively the system at any K < K c since the correlation length diverges at criticality. The geometrical ojects (polymers) involved in the scale adapted expansion live on a suitale scale length l whereas in the usual high temperature expansions they live on scale one. The small parameter is no more K ut rather the ratio etween the correlation length at the given K and the length scale l at which we analyze the system. Of course the smaller is K c K the larger has to e taken the length l. In the context of random ferromagnetic Ising model with ounded interaction, letting q() = Pro(J x,y > ), R +, we prove that there exists a real function q 0 such that the following holds. If for some [0, K c ) we have q() < q 0 (), then the system admits, with proaility one w.r.t. the disorder, a convergent graded cluster expansion implying, in particular, the exponential decay of correlations with a random prefactor [3]. The results in [7] can thus e seen as a special case of the aove statement. We emphasize that since we consider situations aritrarily close to criticality, the first step of our procedure, consisting in the integration over the good region, is a scale adapted expansion. In other words, the minimal length scale involved in the perturative expansion developed in the present paper, is not one as in [7], ut rather depends on the thermodynamic parameters and diverges when approaching the critical point. The multi scale analysis of the ad regions, simpler than the one in [7], is achieved y exploiting the peculiarities of the model. In particular, the asic proaility estimates are deduced via a stochastic domination y a Bernoulli random field. For x = (x 1, x 2 ) R 2 we let x := x 1 + x 2. The spatial structure is modeled y the two dimensional lattice L := Z 2 endowed with the distance D(x, y) := x y. We let e 1 and e 2 e the coordinate unit vectors. As usual for Λ, L we set D(Λ, ) := inf{d(x, y), x Λ, y } and Diam(Λ) := sup{d(x, y), x, y Λ}. The notation Λ L means that Λ is a finite suset of L. We let E := { {x, y} L : D(x, y) = 1 } e the collection of onds in L. Given a positive integer m we let F m e the collection of all the finite susets of L which can e written as disjoint unions of squares with sides of length m parallel to the coordinate axes. 2

4 The single spin state space is X 0 := { 1, +1} which we consider endowed with the discrete topology, the associated Borel σ algera is denoted y F 0. The configuration space in Λ L is defined as X Λ := X0 Λ and considered equipped with the product topology and the corresponding Borel σ algera F Λ. We let X L =: X and F L =: F. Given Λ L and σ := {σ x X {x}, x Λ} X Λ, we denote y σ the restriction of σ to namely, σ := {σ x, x }. Let m e a positive integer and let Λ 1,..., Λ m L e pairwise disjoint susets of L; for each σ k X Λk, with k = 1,..., m, we denote y σ 1 σ 2 σ m the configuration in X Λ1 Λ m such that (σ 1 σ 2 σ m ) Λk = σ k for all k {1,..., m}. A function f : X R is called local iff there exists Λ L such that f F Λ namely, f is F Λ measurale for some ounded set Λ. If f F Λ we shall sometimes misuse the notation y writing f(σ Λ ) for f(σ). We also introduce C(X ) the space of continuous functions on X which ecomes a Banach space under the norm f := sup σ X f(σ) ; note that the local functions are dense in C(X ). We let J := R E, which we consider equipped with its Borel σ algera B. We denote y J e, e E, the canonical coordinates on J. Let P 0 e a proaility measure on R with compact support in R + namely, there exists a real M > 0 such that P 0 ([0, +M]) = 1. We define on (J, B) the product measure P := P E 0. Given Λ L, the disordered finite volume Hamiltonian is the function H Λ : X J R defined as H Λ (σ, J) := J {x,y} σ x σ y (1.2) {x,y} L: {x,y} Λ Given J J, we define the quenched (finite volume) Gis measure in Λ, with oundary condition τ X, as the following proaility measure on X Λ. Given σ X Λ we set where µ τ Λ,J(σ) := Z Λ (τ, J) := 1 Z Λ (τ, J) exp { + H Λ (στ Λ c, J) } (1.3) σ X Λ exp { + H Λ (στ Λ c, J) } (1.4) Note that, for notation convenience, we chaged the signs in the definition of the Hamiltonian (1.2) and in the Gis measure (1.3). Theorem 1.1 Let K c := (1/2) log(1 + 2) e the critical coupling of the standard two dimensional Ising model. There exists a function q 0 : [0, K c ) (0, 1] such that the following holds. Suppose that for some [0, K c ) q q() := 1 P 0 ( [0, ] ) q0 () (1.5) then there exists a positive integer l = l() and a set J B, with P( J ) = 1, such that the following statements hold. There exist two families {Ψ X,Λ, Φ X,Λ : X J R, X L, Λ F l }, called effective potential, such that: Ψ X,Λ, Φ X,Λ F X Λ c B and for each Λ, Λ F l, X L such that X Λ = X Λ one has that Ψ X,Λ = Ψ X,Λ and Φ X,Λ = Φ X,Λ. Moreover for each Λ F l 3

5 1. for each (τ, J) X J we have the convergent expansion log Z Λ (τ, J) = [Ψ X,Λ (τ, J) + Φ X,Λ (τ, J)] (1.6) X Λ 2. for each x L there exists a function r x : J N such that for each J J we have Ψ X,Λ (, J) = 0 for X L such that diam(x) > r x (J) and X x; 3. there exist reals α > 0 and C < such that for any J J sup x L X x e α Diam(X) sup Λ F l Φ X,Λ (, J) < C (1.7) In the deterministic case, J x,y = K with K [0, K c ), the expansion (1.6) holds with Ψ = 0 [14]. In such a case (1.7) implies one of the Dorushin Shlosmann equivalent conditions for complete analyticity, see equation (2.15) in [3]. On the other hand, in our disordered setting the family Ψ does not vanish due to the presence of aritrarily large regions of strong couplings. Nevertheless in item 2 we state that the range of the effective potential Ψ, although unounded, is finite with proaility one. We emphasize that it is not possile to deduce complete analyticity from Theorem 1.1. In [5] we shall prove a similar result in a more general context and, y using the cominatorial approach in [3], deduce an exponential decay of correlations from the convergence of the graded cluster expansion. 2. Graded cluster expansion In this section we prove, relying on some proaility estimate on the multi scale geometry of the disorder which are discussed in Section 3, Theorem 1.1. We follow a classical strategy in disordered systems. Let us fix a realization J J of the random couplings. We first perform a cluster expansion in the regions where the model satisfies a strong mixing condition implying an effective weak interaction on a proper scale. We are then left with an effective residual interaction etween the regions with strong couplings. Since large values of coupling constants have small proaility, the regions of strong couplings are well separated on the lattice; we can thus use the graded cluster expansion developed in [2] to treat the residual interaction Good and ad events Given a positive integer l, we consider the l rescaled lattice L (l) := (lz) 2, which is emedded in L namely, points in L (l) are also points in L, and for each i L (l) we set Q l (i) := {x L : i 1 x 1 i 1 + l 1 and i 2 x 2 i 2 + l 1} (2.1) For i L (l) and [0, K c ), we introduce the ad event E i E (l), i := e E: e Q l (i) { Je > } (2.2) 4

6 Note that E i occurs iff in the square Q l (i) there exists a coupling, taking into account also the oundary onds, larger than. We then define the inary random variale ω i ω (l), i : J {0, 1} as ω i ω (l), i := 1I (l), E (2.3) i Given J J we say that a site i L (l) is good (resp. ad) if and only if ω i (J) = 0 (resp. ω i (J) = 1) and we set L (l) (J) := {i L(l) : ω i (J) = 0} On goodness In this susection we clarify to which extent the good sites in L (l) are good. We shall show that for l large enough, given [0, K c ) and J J, on the good part of the lattice (J) the quenched disordered model satisfies a strong mixing condition allowing a nice cluster expansion. Few more definitions are needed; let i L (l) and k {1, 2}, we denote y P i,k the family of all not empty susets I L (l) such that for each j I we have j k = i k and j h {i h l, i h, i h + l} for h = 1, 2 and h k. We set L (l) I ± := (l) I {j L (l) : j k = i k ± l} where for any I L (l) we have set (l) I := {j L (l) \ I : D(j, I) = l}. For σ X we set σ ± := σ i I± Q l (i) and σ 0 := σ ( i I+ I Q l (i)) c. Moreover for each I L(l) we set O l I := i I Q l(i) and for each X L we set O l X := {i L (l) : X Q l (i) }. Lemma 2.1 Let [0, K c ), there exists an integer l 0 = l 0 () and a real m 0 = m 0 () > 0 such that for each l multiple of l 0, J J, and i L (l) we have Z Ol (I L sup sup sup (l) )(σ +σ τ 0, J)Z Ol (I L (l) )(ζ +ζ τ 0, J) k=1,2 I P i,k σ,ζ,τ X Z Ol (I L (l) )(σ +ζ τ 0, J)Z Ol (I L (l) )(ζ +σ τ 0, J) 1 < e m 0l (2.4) Proof. The proof, which is ased on classical results on (not disordered) two dimensional Ising models adapted to the present not translationally invariant interaction, is organized in three steps. Given [0, K c ) we let J := [0, ] E J. We first prove that for each J J there exists a unique infinite volume Gis measure w.r.t. the local specification µ τ Λ,J, Λ L, τ X. Then we show that the corresponding infinite volume two point correlations decay exponentially with the distance. From this we finally derive the ound (2.4). We consider X endowed with the natural partial ordering σ σ iff for any x L we have σ x σ x. Given two proailities µ, ν on X we write µ ν iff for any continuous increasing (w.r.t. the previous partial ordering) function f we have µ(f) ν(f). Here µ(f) denotes the expectation of f w.r.t. the measure µ. Step 1. Let us denote y + (resp. ) the configuration with all the spins equal to +1 (resp. 1). By monotonicity, which is a consequence of the FKG inequalities, see e.g. Theorem in [8], we get that for each J J and A F lim µ ± Λ,J (A) =: µ± J (A) (2.5) Λ L 5

7 where the limit is taken along an increasing sequence invading L. Moreover, again y the FKG inequalities, we have that any infinite volume Gis measure µ J satisfies the inequalities µ J µ J µ + J (2.6) We now notice that for each J J and x L lim µ ± Λ,J (σ Λ L x) = 0 (2.7) indeed if we let B J e such that B e =, e E, we have that for each J J, x L and Λ L µ Λ,B (σ x) µ Λ,J (σ x) 0 µ + Λ,J (σ x) µ + Λ,B (σ x) (2.8) where we y used the Griffiths inequalities, see e.g. Theorem in [8]. By using [11, 16, 17] and the FKG inequalities we have that (2.8) implies (2.7) since 0 < K c. Again y FKG, equations (2.5) and (2.7) imply that for each J J the infinite volume Gis measure w.r.t. the local specification µ τ Λ,J, Λ L, τ X is unique; we denote this measure y µ J. Step 2. Let x, y L, y the Griffiths inequalities we have that for each J J 0 µ J (σ x ; σ y ) = µ J (σ x σ y ) µ B (σ x σ y ) (2.9) where we recall B has een defined aove (2.8). By using (2.9) and classical exact results on two dimensional Ising model, see e.g. [1], we have that there exists a positive real C 3 () < such that for any x, y L µ J (σ x ; σ y ) µ B (σ x σ y ) C 3 () e D(x,y)/C 3() (2.10) Step 3. We oserve that the argument of the proof of the Theorem 2.1 in [10] applies to the present not translationally invariant setting. Indeed, it depends only on the Leowitz inequalities, which hold true, and the ound (2.10). We thus get that there exists a positive real C 2 () < such that for each τ, τ X, J J, Λ L, Λ, and A F µ τ Λ,J (A) µτ Λ,J (A) C 2() e D(Λc, )/C 2 () (2.11) which is, in the terminology introduced in [12], the weak mixing condition for the local specification µ τ Λ,J. By exploiting the two dimensionality of the model and y using the result in [13], we get that there exist an integer l 1 () and a positive real C 1 () < such that for any τ X, J J, Λ F l1 (), Λ, x L \ Λ, and A F µ τ Λ,J(A) µ τ x Λ,J(A) C 1 () e D(x, )/C 1() (2.12) where τ x X is given y τ x x = τ x and τ x y = τ y for all y x, and we recall that the collection of volumes F l has een defined in Section 1. Again in the terminology of [12], the ound (2.12) is called strong mixing condition. By Corollary 3.2, equations (3.9) and (3.14) in [14], see also equation (2.5.32) in [15], it implies the statement of the Lemma. 6

8 2.3. Cluster expansion in the good region In this susection we cluster expand the partition function over the good part of the lattice. Consider a positive integer l and the l rescaled lattice L (l) = (lz) 2. We denote y D l (i, j) = (1/l)D(i, j) the natural distance in L (l) and y Diam l (I) := sup i,j I D l (i, j) the diameter of a suset I L (l). Given I L (l) and a real r > 0, we denote y B r (l) (I) := {j L (l) : D l (I, j) r} the r neighorhood of I. We associate with each site i L (l) the single site lock spin configuration space X (l) i := X Ql (i). Given I L we consider the lock spin configuration space X (l) I := i I X (l) i, I L (l), equipped with the product topology and the corresponding Borel σ algera F (l) I. As efore we set X (l) := X (l) and F (l) := F (l). L (l) L (l) As for the lattices, see the definition just aove the Lemma 2.1, we introduce operators which allow to pack spins and unpack lock spins. We define the packing operator O l : X X (l) associating with each spin configuration σ X the lock spin configuration O l σ X (l) given y (O l σ) i := {σ x, x Q l (i)}, i L (l). The unpacking operator O l : X (l) X associates with each lock spin configuration ζ X (l) the unique spin configuration O l ζ X such that ζ i = {(O l ζ) x, x Q l (i)} for all i L (l). We remark also that the two operators allow the packing of the spin σ algera and the unpacking of the lock spin one namely, for each I L (l) and Λ L we have ( (l)) O l F I = FOl I and O l( ) (l) F Λ F (2.13) O l Λ Where in the last relation the equality etween the two σ algeras stands if and only if O l O l Λ = Λ. Given L (l) we define the lock spin Hamiltonian H (l) : X (l) J R as H (l) (ζ, J) := H O l (O l ζ, J) for ζ X (l) and J J. The corresponding finite volume Gis measure, with oundary condition ξ X (l), is denoted y µ (l),ξ,j, the partition function y Z (l) (ξ, J) namely, Z (l) (ξ, J) = Z O l (O l ξ, J) (2.14) Let J J, ξ X (l), L (l), and recall L (l) (J) has een defined elow (2.3). In the following Proposition 2.2 we cluster expand Z (l) J) and show, in particular, L (l) (J)(ξ, that Condition 2.1 in [2] is satisfied. To state the result we need few more definitions. Let E (l) := {{x, y} L (l) : D l (x, y) = 1} the collection of edges in L (l). We say that two edges e, e E (l) are connected if and only if e e. A suset (V, E) (L (l), E (l) ) is said to e connected iff for each pair x, y V, with x y, there exists in E a path of connected edges joining them. We agree that if V = 1 then (V, ) is connected. For X L (l) finite we then set T l (X) := inf { E, (V, E) (L (l), E (l) ) is connected and V X } (2.15) Note that T l (X) = 0 if X = 1 and for x, y L (l) we have T l ({x, y}) = D l (x, y). 7

9 Proposition 2.2 Let [0, K c ) and l 0 = l 0 () as in Lemma 2.1. Then for all integer l multiple of l 0, J J, ξ X (l), and L (l) we have log Z (l) J) = V (l) L (l) (J)(ξ, I, (ξ, J) (2.16) for a suitale collection of local functions V (l) satisfying: I (l) := {V I, : X (l) J R, I } 1. given, L (l) if I = I then V (l) (l) I, (, J) = V I, (, J) for any J J ; 2. V (l) (l) I, (, J) F I ( L (l) (J))c for any J J ; 3. if I ( B (l) 6 ( ) ) c then V (l) I, = 0. Moreover, the effective potential V (l) can e ounded as follows. There exist reals α 1 = α 1 () > 0, A 1 = A 1 () <, and n 1 = n 1 () < such that for any J J sup i L (l) I i e α 1l T l (I) sup L (l) : I V (l) I, (, J) A 1 l n 1 (2.17) Proof of Proposition 2.2. The proof can e achieved y applying the arguments in [14,15], where this result is proven with periodic oundary conditions. We refer to Theorem 5.1 in [4] for the modifications needed to cover the case of aritrary oundary conditions and for the stated l dependence of the ound (2.17). Item 3 follows from Figures 2 and 3 in [4] Geometry of adness To characterize the sparseness of the ad region we follow the ideas developed in [2,4,7]. Definition 2.3 We say that two strictly increasing sequences Γ = {Γ k } k 1 and γ = {γ k } k 1 are moderately steep scales iff they satisfy the following conditions: 1. Γ 1 2, and Γ k < γ k /2 for any k 1; 2. for k 1 set ϑ k := 3. k=1 4. a 0 := Γ k γ k 1 2 k (Γ h + γ h ) and λ := inf k 1 (Γ k+1 /ϑ k ), then λ 5; h=1 2 k log[2(γ k+1 + γ k+1 ) + 1] 2 < + ; k=0 8

10 5. for each a > 0 we have [2(ϑ k + Γ k ) + 1] 2 exp{ a2 k 1 } <. k=1 We remark that items 4 and 5 differ slightly from the corresponding ones in Definition 3.1 in [4]. This is due to fact that the analysis of the geometry of ad sets given in the present paper is ased on a stochastic domination argument while the one in 3 in [4] on mixing properties of the disorder. Definition 2.4 We say that G := {G k } k 0, where each G k is a collection of finite susets of L (l), is a graded disintegration of L (l) iff 1. for each g k 0 G k there exists a unique k 0, which is called the grade of g, such that g G k ; 2. the collection k 0 G k of finite susets of L (l) is a partition of the lattice L (l) namely, it is a collection of not empty pairwise disjoint finite susets of L (l) such that g = L (l). (2.18) g G k k 0 Given G 0 L (l) and Γ, γ steep scales, we say that a graded disintegration G is a gentle disintegration of L (l) with respect to G 0, Γ, γ iff the following recursive conditions hold: 3. G 0 = { {i}, i G 0 } ; 4. if g G k then Diam l (g) Γ k for any k 1; 5. set G k := g G k g L (l), B 0 := L (l) \ G 0 and B k := B k 1 \ G k, then for any g G k we have D l (g, B k 1 \ g) > γ k for any k 1; 6. given g G k, let Y 0 (g) := {j L (l) : inf i Q(g) [ i 1 j 1 i 2 j 2 ] ϑ k } where Q(g) L (l) is the smallest rectangle, with axes parallel to the coordinate directions, that contains g; then for each i L (l) we have κ i := sup { k 1 : g G k such that Y 0 (g) i} < We call k gentle, resp. k ad, the sites in G k, resp. B k. The elements of G k, with k 1, are called k gentle atoms. Finally, we set G k := h k G h. We next state a proposition, whose proof is the topic of Section 3, that will ensure that for q small enough the ad sites of the lattice L (l) can e classified according to the notion of gentle disintegration for suitale scales. We note that items 2 and 3 in Definition 2.3 force a super exponential growth of the sequences Γ and γ. It is easy to show that, given β 8, the sequences Γ k := e (β+1)(3/2)k and γ k := 1 8 eβ(3/2)k+1 for k 1 (2.19) 9

11 are moderately steep scales in the sense of Definition 2.3. Given [0, K c ), let α 1 (), A 1 (), and n 1 () as in Proposition 2.2. It is easy to show that there exists β 0 = β 0 () such that the scales Γ, γ in (2.19) with β = β 0 satisfy the conditions stated in items 1 4 in the hypotheses of Theorem 2.5 in [2] for any l large enough. We understand that the constants α and A in those items are to e replaced y α 1 l and A 1 l n 1 respectively. Proposition 2.5 Given [0, K c ) let Γ, γ as in (2.19) with β = β 0 (). There exist q 0 () [0, 1) and a multiple l = l() of l 0 (), see Lemma 2.1, such that if q < q 0 (), recall (1.5), then there exists a B measurale set J J, with P( J ) = 1, such that for each J J there exists a gentle disintegration G(J), see Definition 2.4, of L ( l) with respect to L ( l) (J) and Γ, γ Cluster expansion in the ad region In this susection we sum over the configurations on the ad sites in L (l) \ L (l) (J). We show that provided J is chosen in the full P measure set J J, see Proposition 2.5, it is possile to organize the sum iteratively using a hierarchy of sparse ad regions of the lattice. Proof of Theorem 1.1. We apply Proposition 2.5 to construct the set J and, for each J J, the gentle disintegration G(J) of L ( l) with respect to L ( l) (J) and Γ, γ as in (2.19) with β = β 0 (). For the seek of simplicity we set l = l() in the sequel of the proof. Pick J J, ξ X (l), and L (l) ; y applying Proposition 2.2 we cluster expand log Z (l) J). We get that Condition 2.1 in [2] holds with effective potential V (l) (, J), α = α 1l, A = A 1 l n 1 L (l) (J)(ξ, (here α 1, A 1, and n 1 are the constants appearing in (2.17)), and r = 6. By applying Theorem 2.5 in [2] to the lattice L = L (l) and the gentle disintegration G(J) w.r.t. L (l) (J), Γ, γ, it follows that there exist functions Ψ (l) (l) I, (, J), Φ(l) I, (, J) F I c, with I L(l), such that we have the totally convergent expansion [ ] log Z (l) (ξ, J) = Ψ (l) I, (ξ, J) + Φ(l) I, (ξ, J) (2.20) I Moreover: i) for each, L (l) and each I L (l) such that I = I we have that Ψ (l) I, = Ψ(l) I, and Φ (l) I, = Φ(l) I, ; ii) for J J, for I, L (l), if Diam l (I) > 6 and there exists no g G 1 (J) such that Y 0 (g) = X then Ψ (l) I, (, J) = 0; iii) for each J J we have sup i L (l) I i e cα 1lDiam l (I) [ sup Φ (l) I, (, J) 1 + e cα 1lγ e cα 1 l/4 ] 2 (2.21) L 1 e cα 1l/4 (l) where c = To get the expansion (1.6) we next pull ack the Ψ (l) and Φ (l) to the original scale. We define the family {Ψ X,Λ, Φ X,Λ : X J R, X L, Λ F l } as follows: for each 10

12 τ X, X L, and Λ F l we set { Ψ (l) Ψ X,Λ (τ, J) := I,O l Λ (Ol τ, J) if I L (l) : O l I = X 0 otherwise (2.22) and Φ X,Λ (τ, J) := { Φ (l) I,O l Λ (Ol τ, J) if I L (l) : O l I = X 0 otherwise (2.23) Now, the expansion (1.6) follows from (2.20), (2.14), (2.22), and (2.23). The measuraility properties of the functions Ψ and Φ follow from (2.13) and the analogous properties of the functions Ψ (l) and Φ (l). Item 2 follows from item ii) aove and item 6 in Definition 2.4 once we set r x (J) := l[(γ κx + 2θ κx ) 6] for each x L and J J, where κ x is defined in item 6 of Definition 2.4. We finally prove item 3. Let J J and set α := cα 1, we have sup x L X x By setting e α Diam(X) sup Λ F l Φ X,Λ (, J) = sup x L = sup x L e cα1ldiam(x)/l sup Φ X,Λ (, J) Λ F X x l e cα 1lDiam l (I) sup Φ (l) I, (, J) L (l) I L (l) : O l I x C := 1 + e 4α [ 1 + e α/4 1 e α/4 ] d the ound (1.7) follows y using (2.21) and γ 1 4, see item 1 in Definition 2.3. (2.24) 3. Graded geometry In this section we prove Proposition 2.5. Recalling the random field ω has een introduced in (2.3), we define Ω := {0, 1} L(l) and let A e the corresponding Borel σ algera. We denote y Q = Q (l),, a proaility on Ω, the distriution of the random field ω. Given I L (l) we set A I := σ{ω i, i I} A. Since we assumed the coupling J e, e E, to e i.i.d. random variales, the measure Q is translationally invariant. Let us introduce the parameter p which measures the strength of the disorder p := ess sup Q ( ω 0 = 1 A {0} c) (ω) (3.1) ω Ω where the essential supremum is taken w.r.t. Q. We shall first prove that if p is small enough, depending on the parameter a 0 appearing in item 4 in the Definition 2.3, then we can construct a gentle disintegration in the sense of Definition 2.4. We finally show that the aove condition is met if q 0 () in (1.5) is properly chosen. Let us first descrie an algorithm to construct the family G introduced in Definition 2.4. Given a configuration ω Ω and Γ, γ moderately steep scales, we define the following 11

13 inductive procedure in a finite volume Λ L (l) which finds the k gentle sites in Λ. Set G 0 := L (l) (ω), G 0 := {{i}, i G 0 }, and B 0 := L (l) \ G 0. At step k 1 do the following: 1. i = 1 and V = ; 2. if (B k 1 Λ) \ V = then goto 6; 3. pick a point x (B k 1 Λ) \ V. Set A = B (l) Γ k (x) B k 1 and V = V A; 4. if Diam l (A) Γ k and D l (A, B k 1 \ A) > γ k then gk i = A and i = i + 1; 5. goto 2; 6. set G k := {g m k, m = 1,..., i 1}, with the convention G k = if i = 1, G k := i 1 m=1 gm k, and B k := B k 1 \ G k. Set now k = k + 1 and repeat the algorithm until Γ k > Diam l Λ. Let us riefly descrie what the aove algorithm does. At step k we have inductively constructed B k 1, the set of (k 1) ad sites; we stress that sites in L (l) \ Λ may elong to B k 1. Among the sites in B k 1 Λ we are now looking for the k gentle ones. The set V is used to keep track of the sites tested against k gentleness. At step 3 we pick a new site x B k 1 Λ and test it, at step 4, for k gentleness against B k 1, i.e. including also ad sites in L (l) \ Λ. Note that the families G k for any k 1 are independent on the way in which x is chosen at step 3 of the algorithm. Suppose, indeed, to choose x (B k 1 Λ)\V at step 3 and to find that A = B (l) Γ k (x) B k 1 is a k gentle cluster. Consider x A such that x x and set A := B (l) Γ k (x ) B k 1 : since A passes the test against k gentleness at step 4 of the algorithm, we have A A. By changing the role of x and x we get A = A. After a finite numer of operations (ounded y a function of Λ ), the algorithm stops and outputs the family G k (Λ) (note we wrote explicitly the dependence on Λ) with the following property. If g G k (Λ) then Diam l (g) Γ k and D l (g, B k 1 (Λ) \ g) > γ k. Note that g is not necessarily connected. We finally take an increasing sequence of sets Λ i L (l), invading L (l) and we sequentially perform the aove algorithm. This means the algorithm for Λ i is performed independently of the outputs previously otained. It is easy to show that if g G k (Λ i ) then g G k (Λ i+1 ); therefore G k (Λ i ) is increasing in i 1, so that we can define G k := lim i G k (Λ i ) = i G k(λ i ) and G k := lim i G k (Λ i ) = g G k g. Hence, B k (Λ i ) = B k 1 (Λ i ) \ G k (Λ i ) = L (l) \ k 1 j=0 G j(λ i ) is decreasing in i 1, so that B k := lim i B k (Λ i ) = i B k(λ i ). We also remark that, y construction, {B k, k 0} is a decreasing sequence. Note that from the construction it follows that it is possile to decide whether a site x is k gentle y looking only at the ω s inside a cue centered at x of radius ϑ k, as defined in item 2 of Definition 2.3. Hence, see Lemma 3.4 in [4], we have the following lemma. Lemma 3.1 Let G k and G k, k = 0, 1,..., as constructed aove. Then for each x L (l) {ω : x G k (ω)} A B (l) ϑ k (x) (3.2) 12

14 Theorem 3.2 Let the sequences Γ, γ satisfy the conditions in items 1, 2, and 4 in Definition 2.3. Let also p < exp{ a 0 /2} and set a := log p a 0 /2 > 0. Then Q (x B k ) exp{ a 2 k } (3.3) Remark. From the previous ound and item 5 in Definition 2.3, via a straightforward application of Borel Cantelli lemma, see the proof of Theorem 3.3 in [4] for the details, we deduce the following. There exists an A measurale set Ω Ω, with Q( Ω) = 1, such that for each ω Ω there exists a gentle disintegration G(ω), see Definition 2.4, of L (l) with respect to L (l) (ω) and Γ, γ. The first step in proving Theorem 3.2 consists in replacing the not product measure Q y a Bernoulli product measure with parameter p. This is a standard argument that we report for completeness. We consider Ω endowed with the natural partial ordering ω ω iff for any x L (l) we have ω x ω x. Given two proailities Q, P on Ω we write Q P iff for any continuous increasing (w.r.t. the previous partial ordering) function f we have Q(f) P (f). Lemma 3.3 Let Q p e the Bernoulli measure on Ω with marginals Q p (ω x = 1) = p and recall the parameter p has een defined in (3.1). Then Q Q p. Proof. For Λ L (l) we denote y Q Λ the marginal of Q on Ω Λ = {0, 1} Λ and y Q p the Bernoulli measure on {0, 1}, Q p ({1}) = p. The lemma follows y induction from It is easy to show Q Λ (dω Λ ) Q p (dω x )Q Λ\{x} (dω Λ\{x} ) x Λ, Λ L (l) (3.4) Q Λ ( ωx = 1 AΛ\{x} ) = QΛ c therefore (3.1) and the translation invariance of Q imply ( Q ( ω x = 1 A{x} c) ) Q Λ a.s. ess sup ( Q Λ ωx = 1 )( ) AΛ\{x} ωλ\{x} p (3.5) ω Λ\{x} We next prove (3.4). Let f e a continuous and increasing function on Ω Λ ; y taking conditional expectation we have ( [ ] ) Q Λ (f) = Q Λ\{x} (Q ) Λ f 1I{ωx=1} + 1I AΛ\{x} {ωx=0} ( ) { = Q Λ\{x} dωλ\{x} f ( ) ω Λ\{x} 0 {x} ( + Q Λ ωx = 1 (ω) )[ ( ) ( )] } F Λ\{x} f ωλ\{x} 1 {x} f ωλ\{x} 0 {x} ( ) { Q Λ\{x} dωλ\{x} p [ f ( ) ( )] ( ) } ω Λ\{x} 1 {x} f ωλ\{x} 0 {x} + f ωλ\{x} 0 {x} ( ) = Q Λ\{x} dωλ\{x} Qp (dω x )f(ω Λ ) where we used that f is increasing and (3.5) in the inequality. (3.6) 13

15 Lemma 3.4 For each x L (l) and k 0 the event {ω : x B k (ω)} is increasing namely, ω ω = B k (ω) B k (ω ) (3.7) Proof. We prove (3.7) y induction on k. First of all we note that y definition of the natural partial order on Ω it holds for k = 0. Let us prove that G k+1 (ω ) k+1 j=0 G j (ω) (3.8) Let x G k+1 (ω ), then either x k j=0 G j(ω) or x B k (ω). In the former case we are done, in the latter we have that, since x G k+1 (ω ), there exists a set g B k (ω ) such that: i) x g ; ii) Diam l (g ) Γ k ; iii) D l (g, B k (ω ) \ g ) > γ k. Set now g := g B k (ω), y the inductive hypotheses it is easy to verify that g satisfies the three properties aove with ω replaced y ω. Hence x G k+1 (ω). From (3.8) and the induction hypotheses we get k+1 j=0 G j(ω ) k+1 j=0 G j(ω). Since B k+1 (ω) = B 0 (ω) \ k+1 j=0 G j(ω), we have proven (3.7) with k replaced y k + 1. The key step in proving Theorem 3.2 is the following recursive estimate on the degree of adness. Lemma 3.5 Let Γ, γ satisfy the conditions in items 1, 2, and 4 in Definition 2.3 and set ψ k := Q p (x B k ), note ψ k is independent of x y translational invariance, and A k (x) := B (l) γ k +Γ k (x) \ B (Γk 1)/2(x). Then ψ k+1 A k+1 ψ 2 k (3.9) where A k = A k (x) does not depend on x. Proof. By recalling the definition of the k ad set B k we have {x B k+1 } = {x B k } {x G k+1 } (3.10) On the other hand, y the construction of the (k + 1) gentle sites, {x B k } {x G k+1 } {x B k } { y A k+1 (x) : y B k } (3.11) indeed, given B k, if there were no k ad site in the annulus A k+1 (x) then x would have een (k + 1) gentle. From (3.10) and (3.11) ( ) ψ k+1 = Q p (x B k+1 ) Q p {x B k } {y B k } = y A k+1 (x) y A k+1 (x) y A k+1 (x) Q p ( {x Bk } {y B k } ) Q p ( {x Bk } ) Q p ( {y Bk } ) = A k+1 ψ 2 k 14 (3.12)

16 where in the last step, we used (3.2), the definition of A k (x), item 2 in Definition 2.3, the product structure of the measure Q p and its translation invariance. Proof of Theorem 3.2. By Lemmata 3.3 and 3.4 it is enough to prove the ound (3.3) for the Bernoulli measure Q p. Let f k := log ψ k and k := log A k+1, where ψ k and A k have een defined in Lemma 3.5. Then y iterating (3.9) and using item 4 in Definition 2.3, we get f k+1 2f k k 2 k+1 f 0 2 k k 2 j j 2 k+1 f 0 2 k a 0 = 2 k+1 a (3.13) j=0 where we recall that a = log p a 0 /2 = f 0 a 0 /2 > 0. Recall q has een defined in (1.5). To gently disintegrate the lattice L (l) y means of Theorem 3.2, we need a ound of the adness parameter p, see (3.1), in terms of q. Lemma 3.6 Recall p has een defined in (3.1) and q in (1.5); then p 1 (1 q) 2l(l 1) 1 (1 q) l + 4 (3.14) 1 (1 q) 2l(l+1) Proof. For each i L (l) we define the five events Ei 0, E1,± i, and E 2,± i : E 0 i := e E: e Q l (i) { Je > } and E s,± i := e E: e Q l (i), e Q l (i±les) { Je > } (3.15) where s = 1, 2 and we recall e 1 and e 2 are the coordinate unit vectors in L. By using the equality E i = Ei 0 E1, i E 2,+ i E 1,+ i E 2, i and the product nature of P, we have that P(E i {ω j = a j } j i ) P(Ei 0 {ω j = a j } j i ) 2 + P(E s,+ i {ω j = a j } j i ) + s=1 2 s=1 = P(Ei 0) + 4P(E1, i ω i le1 = a i le1 ) P(E s, i {ω j = a j } j i ) (3.16) Since E 1, i {ω i le1 = 0} =, the l.h.s. of (3.16) can e ounded uniformly in {a j } j i y P(Ei 0 ) + 4P(E 1, i )/P(ω i le1 = 1). The lemma then follows y a straightforward computation. Proof of Proposition 2.5. Let the functions q 0 : [0, K c ) (0, 1] and l : [0, K c ) [0, ) e constructed as follows. 1. By Lemma 2.1, given [0, K c ), we find l 0 () and m 0 () such that (2.4) holds. 15

17 2. By Proposition 2.2 we find α 1 (), A 1 (), and n 1 () such that the ound (2.17) holds. 3. Choose β 0 () as elow (2.19) and let the scales Γ, γ e as in (2.19) with β = β 0 (). 4. Compute a 0 () in item 4 in Definition Let q 0 () and l(), with l() multiple of l 0 () such that for any q q 0 (). 1 (1 q) 2 l()( l() 1) 1 (1 q) l() (1 q) < exp{ a 0()/2} 2 l()( l()+1) Step 5 is possile ecause for each l() fixed the l.h.s. of the inequality aove converges to 2/(l + 1) as q 0. By applying Lemma 3.6, Theorem 3.2, and the remark following it we conclude the proof of the proposition. References [1] R.J. Baxter, Exactly solved models in Statistical Mechanics. London: Academic Press, [2] L. Bertini, E.N.M. Cirillo, E. Olivieri, Graded cluster expansion for lattice systems. Comm. Math. Phys. 258, (2005). [3] L. Bertini, E.N.M. Cirillo, E. Olivieri, A cominatorial proof of tree decay of semi invariants, J. Statist. Phys. 115, (2004). [4] L. Bertini, E.N.M. Cirillo, E. Olivieri, Renormalization group in the uniqueness region: weak Gisianity and convergence. Comm. Math. Phys. 261, (2006). [5] L. Bertini, E.N.M. Cirillo, E. Olivieri, in preparation. [6] S.F. Edwards, P.W. Anderson, Theory of spin glasses. J. Phys. F Metal Phys. 5, (1975). [7] J. Fröhlich, J.Z. Imrie, Improved perturation expansion for disordered systems: eating Griffiths singularities. Comm. Math. Phys. 96, (1984). [8] J. Glimm, A. Jaffe, Quantum physics. A functional integral point of view. Second edition. New York: Springer Verlag, [9] R.B. Griffiths, Non analityc ehavior aove the critical point in a random Ising ferromagnet. Phys. Rev. Lett. 23, (1969). [10] Y. Higuchi, Coexistence of infinite ( )-clusters. II. Ising percolation in two dimensions. Proa. Theory Related Fields 97, 1 33 (1993). 16

18 [11] J.L. Leowitz, A. Martin Löf, On the uniqueness of the equilirium state for ising spin system. Comm. Math. Phys. 25, (1972). [12] F. Martinelli, E. Olivieri, Approach to equilirium of Glauer dynamics in the one phase region I. The attractive case. Commun. Math. Phys. 161, (1994). [13] F. Martinelli, E. Olivieri, R. Schonmann, For 2 D lattice spin systems weak mixing implies strong mixing. Commun. Math. Phys. 165, (1994). [14] E. Olivieri, On a cluster expansion for lattice spin systems: a finite size condition for the convergence. J. Statist. Phys. 50, (1988). [15] E. Olivieri, P. Picco, Cluster expansion for D dimensional lattice systems and finite volume factorization properties. J. Statist. Phys. 59, (1990). [16] L. Onsager, Crystal statistics I. A two dimensional model with an order disorder transition. Phys. Rev. 65, (1944). [17] D. Ruelle, On the use of small external field in the prolem of symmetry reakdown in statistical mechanics. Ann. Phys. 69, (1972). [18] A. Suto, Weak singularity and asence of metastaility in random Ising ferromagnets. J. Phys. A 15, L7494 L752 (1982). 17