A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model

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1 A new proof of the sharpness of the phase transition for Bernoulli percolation an the Ising moel Hugo Duminil-Copin an Vincent Tassion January 20, 208 Abstract We provie a new proof of the sharpness of the phase transition for Bernoulli percolation an the Ising moel. The proof applies to infiniterange moels on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime β < β c, an the mean-fiel lower boun P β [0 ] (β β c )/β for β > β c. For finite-range moels, we also prove that for any β < β c, the probability of an open path from the origin to istance n ecays exponentially fast in n. For the Ising moel, we prove finiteness of the susceptibility for β < β c, an the mean-fiel lower boun σ 0 + β (β 2 βc 2 )/β 2 for β > β c. For finite-range moels, we also prove that the two-point correlation functions ecay exponentially fast in the istance for β < β c. The paper is organize in two sections, one evote to Bernoulli percolation, an one to the Ising moel. While both proofs are completely inepenent, we wish to emphasize the strong analogy between the two strategies. General notation. Let G = (V, E) be a locally finite (vertex-)transitive infinite graph, together with a fixe origin 0 V. For n 0, let Λ n = {x V (x, 0) n}, where (, ) is the graph istance. Consier a set of coupling constants (J x,y ) x,y V with J x,y = J y,x 0 for every x an y in V. We assume that the coupling constants are invariant with respect to some transitively acting group. More precisely, there exists a group Γ of automorphisms acting transitively on V such that J γ(x),γ(y) = J x,y for all γ Γ. We say that (J x,y ) x,y V is finite-range if there exists R > 0 such that J x,y = 0 whenever (x, y) > R. Bernoulli percolation. The main result Let P β be the bon percolation measure on G efine as follows: for x, y V, {x, y} is open with probability e βjx,y, an close with probability e βjx,y.

2 We say that x an y are connecte in S V if there exists a sequence of vertices (v k ) 0 k K in S such that v 0 = x, v K = y, an {v k, v k+ } is open for every 0 k < K. We enote this event by x S y. For A V, we write x S A for the event that x is connecte in S to a vertex in A. If S = V, we rop it from the notation. Finally, we set 0 if 0 is connecte to Λ c n for all n. The critical parameter is efine by β c = inf{β 0 P β [0 ] > 0}. Theorem... For β > β c, P β [0 ] β βc β. 2. For β < β c, the susceptibility is finite, i.e. P β [0 x] <. x V 3. If (J x,y ) x,y V is finite-range, then for any β < β c, there exists c = c(β) > 0 such that P β [0 Λ c n] e cn for all n 0. Let us escribe the proof quickly. For β > 0 an a finite subset S of V, efine ϕ β (S) = ( e βjx,y )P β (0 S x). (.) x S y S This quantity can be interprete as the expecte number of open eges on the external bounary of S that are connecte to 0 by an open path of vertices in S. Also introuce β c = sup{β 0 ϕ β (S) < for some finite S V containing 0}. (.2) In orer to prove Theorem., we show that Items, 2 an 3 hol with β c in place of β c. This irectly implies that β c = β c, an thus Theorem.. The quantity ϕ β (S) appears naturally when ifferentiating the probability P β [0 Λ c n] with respect to β. Inee, a simple computation presente in Lemma.4 provies the following ifferential inequality β P β[0 Λ c n] β inf S Λ n 0 S ϕ β (S) ( P β [0 Λ c n]). (.3) By integrating (.3) between β c an β > β c an then letting n ten to infinity, we obtain P β [0 ] β β c β. Now consier β < β c. The existence of a finite set S containing the origin such that ϕ β (S) <, together with the BK-inequality, imply that the expecte size of the cluster the origin is finite. 2

3 .2 Comments an consequences Bibliographical comments. Theorem. was first prove in [AB87] an [Men86] for Bernoulli percolation on the -imensional hypercubic lattice. The proof was extene to general quasi-transitive graphs in [AV08]. The first item was prove in [CC87]. Nearest-neighbor percolation. We recover easily the stanar results for nearestneighbor moel by setting J x,y = 0 if {x, y} E, J x,y = if {x, y} E, an p = e β. In this context, one can obtain the inequality P p [0 ] p p c p( p c) for p p c by introucing ϕ p (S) = p x S y S {x,y} E P p [0 S x]. This lower boun is slightly better than Item of Theorem. an is provie by little moifications in our proof (see [DT5] for a presentation of the proof in this context). Site percolation. As in [AB87], the proof may be aapte to site percolation on transitive graphs. In this context, one can obtain the inequality P p [0 ] p p c ( is the egree of G) for p p c by introucing p c ϕ p (S) = x S y S {x,y} E P p [0 S x]. Finite susceptibility against exponential ecay. Finite susceptibility oes not always imply exponential ecay of correlations for infinite-range moels. Conversely, on graphs with exponential growth, exponential ecay oes not imply finite susceptibility. Hence, in general, the secon conition of Theorem. is neither weaker nor stronger than the thir one. Percolation on the square lattice. On the square lattice, the inequality p c /2 was first obtaine by Harris in [Har60] (see also the short proof of Zhang presente in [Gri99]). The other inequality p c /2 was first prove by Kesten in [Kes80] using a elicate geometric construction involving crossing events. Since then, many other proofs invoking exponential ecay in the subcritical phase (see [Gri99]) or sharp threshol arguments (see e.g. [BR06]) have been foun. Here, Theorem. provies a short proof of exponential ecay an therefore a short alternative to these proofs. For completeness, let us sketch how exponential ecay implies that p c /2: item 3 implies that for p < p c the probability of an open path from left to right in a n by n square tens to 0 as n goes to infinity. But self-uality implies that this oes not happen when p = /2, thus implying that p c /2. Lower boun on β c. Since ϕ β ({0}) = y V e βj 0,y, we obtain a lower boun on β c by taking the solution of the equation y V e βj 0,y =. 3

4 Behaviour at β c. Uner the hypothesis that y V J 0,y <, the set {β 0 ϕ β (S) < for some finite S V containing 0 } efining β c in Equation (.2) is open. In particular, we have that at β = β c = β c, ϕ β (S) for every finite S 0. This implies the following classical result. Proposition.2 ([AN84]). We have P βc [0 x] =. x V Proof. Simply write ( e βcj 0,y ) P βc [0 x] ϕ β (Λ n ) =. y V x V n Semi-continuity of β c. Consier the nearest-neighbor moel. Since β c is efine in terms of finite sets, one can see that β c is lower semi-continuous when seen as a function of the graph in the following sense. Let G be an infinite locally finite transitive graph. Let (G n ) be a sequence of infinite locally finite transitive graphs such that the balls of raius n aroun the origin in G n an G are the same. Then, lim inf β c (G n ) β c (G). (.4) The equality β c = β c implies that the semi-continuity (.4) also hols for β c (this also followe from [Ham57] an the exponential ecay in subcritical, but the efinition of β c illustrates this property reaily). The locality conjecture, ue to Schramm an presente in [BNP], states that for any ε > 0, the map G β c (G) shoul be continuous on the set of graphs with β c < ε. The iscussion above shows that the har part in the locality conjecture is the upper semi-continuity. Depenent moels. For epenent percolation moels, the proof oes not exten in a trivial way, mostly ue to the fact that the BK inequality is not available in general. Nevertheless, this new strategy may be of some use. For instance, for ranom-cluster moels on the square lattice, a proof (see [DST5]) base on the strategy of this paper an the parafermionic observable offers an alternative to the stanar proof of [BD2a] base on sharp threshol theorems. Oriente percolation. The proof applies mutatis mutanis to oriente percolation. Percolation with a magnetic fiel. In [AB87], the authors consier a percolation moel with magnetic fiel efine as follows. A a ghost vertex g V an consier that {x, g} is open with probability e h, inepenently for any x V. Let P β,h be the measure obtaine from P β by aing 4

5 the eges {x, g}. An important results in [AB87] is the following meanfiel lower boun which is instrumental in the stuy of percolation in high imensions (see e.g. [AN84]). Proposition.3 ([AB87]). There exists a constant c > 0 such that for any h > 0, P βc,h[0 g] c h. In Section.5, we provie a short proof of this proposition, using the same strategy as in our proof of Theorem...3 Proof of Item In this section, we prove that for every β β c, Let us start by the following lemma. Lemma.4. Let β > 0 an Λ V finite, P β [0 ] β β c. (.5) β β P β[0 Λ c ] β inf ϕ β (S) ( P β [0 Λ c ]). (.6) S Λ 0 S Before proving this lemma, let us see how it implies (.5). By setting f(β) = P β [0 Λ c ] in (.6), an observing that ϕ β (S) for any β > β c, we obtain the following ifferential inequality: f (β) f(β) β, for β ( β c, ). (.7) Integrating (.7) between β c an β implies that P β [0 Λ c ] = f(β) β β c β every Λ V. By letting Λ ten to V, we obtain (.5). for Proof of Lemma.4. Let β > 0 an Λ. Define the following ranom subset of Λ: S = {x Λ such that x / Λ c }. Recall that {x, y} is pivotal for the configuration ω an the event {0 Λ c } if ω {x,y} {0 Λ c } an ω {x,y} {0 Λ c }. (The configuration ω {x,y}, resp. ω {x,y}, coincies with ω except that the ege {x, y} is close, resp. open.) Russo s formula ([Rus78] or [Gri99, Section 2.4]) implies that β P β[0 Λ c ] = J x,y P β [{x, y} pivotal] (.8) {x,y} β β S 0 ( e βjx,y )P β [{x, y} pivotal an 0 / Λ c ] {x,y} ( e βjx,y )P β [{x, y} pivotal an S = S]. {x,y} 5

6 In the secon line, we use the inequality t e t for t 0. Observe that the event that {x, y} is pivotal an S = S is nonempty only if x S an y S, or y S an x S. Furthermore, the vertex in S must be connecte to 0 in S. We can assume without loss of generality that x S an y S. Rewrite the event that {x, y} is pivotal an S = S as {0 S x} {S = S}. Since the event {S = S} an {0 S x} are measurable with respect to the state of eges having one enpoint in V S, an eges having both enpoints in S respectively. Therefore, the two events above are inepenent. Thus, P β [{x, y} pivotal an S = S] = P β [0 S x]p β [S = S]. Plugging this equality in the computation above, we obtain The proof follows reaily since β P β[0 Λ c ] β ϕ β (S)P β [S = S] S 0 β ( inf ϕ β(s)) P β [S = S]. S 0 S 0 P β [S = S] = P β [0 S ] = P β [0 / Λ c ] = P β [0 Λ c ]. S 0 Remark.. In the proof above, Russo s formula is possibly use in infinite volume, since the moel can be infinite-range. There is no ifficulty resolving this technical issue (which oes not occur for finite-range) by finite volume approximation. The same remark applies below when we use the BK inequality..4 Proof of Items 2 an 3 In this section, we show that Items 2 an 3 in Theorem. hol with β c in place of β c. Lemma.5. Let β > 0, an u S A an B S =. We have P β [u A B] x S y S ( e βjx,y )P β [u S x]p β [y A B]. Proof of Lemma.5. Let u S an assume that the event u A B hols. Consier an open path (v k ) 0 k K from u to B. Since B S =, one can efine the first k such that v k+ S. We obtain that the following events occur isjointly (see [Gri99, Section 2.3] for a efinition of isjoint occurrence): u is connecte to v k in S, {v k, v k+ } is open, v k+ is connecte to B in A. The lemma is then a irect consequence of the BK inequality applie twice (v k plays the role of x, an v k+ of y). 6

7 Let us now prove the secon item of Theorem.. Fix β < β c an S such that ϕ β (S) <. For Λ V finite, introuce χ(λ, β) = max { P β [u Λ v] ; u Λ}. v Λ For every u, let S u be the image of S by a fixe automorphism sening 0 to u. Lemma.5 implies that for every v Λ S u, P β [u Λ v] x S u Summing over all v Λ S u, we fin P β [u Su x]( e βjx,y )P β [y y S u P β [u Λ v] ϕ β (S)χ(Λ, β). v Λ S u Using the trivial boun P β [u v] for v Λ S u, we obtain Optimizing over u, we euce that which implies in particular that P β [u Λ v] S + ϕ β (S)χ(Λ, β). v Λ χ(λ, β) S ϕ β (S) P β [0 Λ S x] x Λ ϕ β (S). The result follows by taking the limit as Λ tens to V. Λ v]. We now turn to the proof of the thir item of Theorem.. A similar proof was use in [Ham57]. Let R be the range of the (J x,y ) x,y V, an let L be such that S Λ L R. Lemma.5 implies that for n L, P β [0 Λ c n] ( e βjx,y )P β [0 S x]p β [y Λ c n] x S y S ϕ β (S)P β [0 Λ c n L]. In the last line, we use that y is connecte to istance larger than or equal to n L since e βjx,y = 0 if x S an y is not in Λ L. By iterating, this immeiately implies that P β [0 Λ c n] ϕ β (S) n/l. 7

8 .5 Proof of Proposition.3 Let us introuce M(β, h) = P β,h [0 g]. Lemma.6 ([AB87]). M β ( J 0,x ) M M x V h. (.9) Proof. Consier a finite subset Λ of V. Russo s formula leas to the following version of (.8): P β,h [0 Λ c {g}] β = J x,y P β,h [{x, y} pivotal]. {x,y} The ege {x, y} is pivotal if, without using {x, y}, one of the two vertices is connecte to 0 but not to Λ c {g}, an the other one to Λ c {g}. Without loss of generality, let us assume that x is connecte to 0, an y is not. Conitioning on the set S = {z Λ z Λ c {g} without using {x, y}}, we obtain P β,h [{x, y} pivotal] P β,h [y Λ c {g}] P β,h [0 x, 0 / Λ c {g}]. Plugging this inequality in (.8) an letting Λ ten to V, we fin M β ( J 0,y ) M ( P β,h [0 x, 0 / g]). {0,y} x V We conclue by observing that if C enotes the cluster of 0 in V, we fin M h = h ( = n=0 x V n=0 P β,h [ C = n]e nh ) = np β,h [ C = n]e nh n=0 P β,h [0 x, C = n]e nh = P β,h [0 x, 0 / g]. x V Another ifferential inequality, which is harer to obtain, usually complements (.9): M h M h + M 2 + βm M β. (.0) This other inequality may be avoie using the following observation. The ifferential inequality (.6) is satisfie with P β [0 Λ c ] replace by P β,h [0 {g} Λ c ], thus giving us for β β c an h 0, M β β ( M) 8

9 (at β = β c, we use the fact that ϕ βc (S) for every finite S 0, see the comment before Proposition.2). When β β c this implies that M β M β β ( x V J 0,x ) M M h, (.) which immeiately implies the following mean-fiel lower boun: there exists a constant c > 0 such that for any h > 0, P βc,h[0 g] = M(β c, h) c h. Remark.2. While (.) is slightly shorter to obtain that (.0), the later is very useful when trying to obtain an upper boun on M(β c, h). 2 The Ising moel 2. The main result For a finite subset Λ of V, consier a spin configuration σ = (σ x x Λ) {, } Λ. For β > 0 an h R, introuce the Hamiltonian H Λ,β,h (σ) = β J x,y σ x σ y x,y Λ h σ x. x Λ Define the Gibbs measures on Λ with free bounary conitions, inverse-temperature β an external fiel h R by the formula f Λ,β,h = f(σ) exp[ H Λ,β,h (σ)] σ {,} Λ exp[ βh Λ,β,h (σ)] σ {,} Λ for f {, } Λ R. Let the infinite-volume Gibbs measure β,h be the weak limit of Λ,β,h as Λ V. Also write + β for the weak limit of β,h as h 0. Introuce β c = inf{β > 0 σ 0 + β > 0}. Theorem 2... For β > β c, σ 0 + β β 2 β 2 c β For β < β c, the susceptibility is finite, i.e. σ 0 σ x + β <. x V 3. If (J x,y ) x,y V is finite-range, then for any β < β c, there exists c = c(β) > 0 such that σ 0 σ x + β e c(0,x) for all x V. This theorem was first prove in [ABF87] for the Ising moel on the - imensional hypercubic lattice. The proof presente here improves the constant in the mean-fiel lower boun, an extens to general transitive graphs. 9

10 The proof of Theorem 2. follows closely the proof for percolation. For β > 0 an a finite subset S of V, efine ϕ S (β) = x S y V S tanh(βj x,y ) σ 0 σ x S,β,0, which bears a resemblance to (.). Similarly to (.2), set β c = sup{β 0 ϕ β (S) < for some finite S V containing 0}. In orer to prove Theorem 2., we show that Items, 2 an 3 hol with β c in place of β c. This irectly implies that β c = β c, an thus Theorem 2.. The proof of Theorem 2. procees in two steps. As for percolation, the quantity ϕ β (S) appears naturally in the erivative of a finite-volume approximation of σ 0 β,h. Roughly speaking (see Lemma 2.6 for a precise statement), one obtains a finite-volume version of the following inequality: β σ 0 2 β,h 2 β inf S 0 ϕ β(s) ( σ 0 2 β,h ). This inequality implies, for every β > β c, σ 0 β,h β 2 β 2 c β 2 an therefore Item by letting h ten to 0. The remaining items follow from an improve Simon s inequality, prove below. Remark 2.. The proof uses the ranom-current representation. In this context, the erivative of σ 0 2 β,h has an interpretation which is very close to the ifferential inequality (.6). In some sense, percolation is replace by the trace of the sum of two inepenent ranom sourceless currents. Furthermore, the strong Simon s inequality plays the role of the BK inequality for percolation. 2.2 Comments an consequences. The ranom-cluster moel (also calle Fortuin-Kasteleyn percolation) with cluster weight q = 2 is naturally couple to the Ising moel (see [Gri06] for etails). The previous theorem implies exponential ecay in the subcritical phase for this moel. 2. Exactly like in the case of Bernoulli percolation, the critical parameter of the ranom-cluster moel on the square lattice with q = 2 can be prove to be equal to 2/( + 2) using the exponential ecay in the subcritical phase together with the self-uality. 3. The previous item together with the coupling with the Ising moel implies that β c = 2 log( + 2) on the square lattice (see [Ons44, BD2b] for alternative proofs). 0

11 4. Exactly as for Bernoulli percolation, we get that ϕ βc (S) for any finite set S 0, which implies the following classical proposition. Proposition 2.2 ([Sim80]). We have σ 0 σ x + β c =. x V Proof. Use Griffiths inequality (2.) below to show that for x Λ n Λ n, σ 0 σ x + β c σ 0 σ x βc,0 σ 0 σ x Λn,β c,0 so that ( y V tanh(βj 0,y )) σ 0 σ x + β c ϕ βc (Λ n ) =. x V n 5. The equality β c = β c implies that β c is lower semi-continuous with respect to the graph (see the iscussion for Bernoulli percolation). 6. In [ABF87], the authors also prove the following result. Proposition 2.3 ([ABF87]). There exists a constant c > 0 such that for any h > 0, σ 0 βc,h ch /3. We present in Section 2.6 a short proof of this proposition, using the same strategy as in our proof of Proposition Preliminaries Griffiths inequality. The following is a stanar consequence of the secon Griffiths inequality [Gri67]: for β > 0, h 0 an S Λ two finite subsets of V, σ 0 S,β,h σ 0 Λ,β,h. (2.) Ranom-current representation. This section presents a few basic facts on the ranom-current representation. We refer to [Aiz82, AF86, ABF87] for etails on this representation. Let Λ be a finite subset of V an S Λ. We consier an aitional vertex g not in V, calle the ghost vertex, an write P 2 (S {g}) for the set of pairs {x, y}, x, y S {g}. We also efine J x,g = h/β for every x Λ. Definition 2.4. A current n on S (also calle a current configuration) is a function from P 2 (S {g}) to {0,, 2,...}. A source of n = (n x,y {x, y} P 2 (S {g})) is a vertex x S {g} for which y S n x,y is o. The set of sources of n n y) is enote by n. We say that x an y are connecte in n (enote by x if there exists a sequence of vertices v 0, v,..., v K in S {g} such that v 0 = x, v K = y an n vk,v k+ > 0 for every 0 k < K.

12 For a finite subset Λ of V an a current n on Λ, efine w(n) = w(n, β, h) = {x,y} P 2 (Λ {g}) (βj x,y ) nx,y. n x,y! From now on, we will write n=a for the sum running on currents on S with sources A. Sometimes, the current n will be on S S (an therefore the sum will run on such currents), but this will be clear from context. An important property of ranom currents is the following: for every subset A of Λ, we have n=a w(n) if A is even, n= w(n) σ a Λ,β,h = a A n=a {g} w(n) if A is o. n= w(n) We will use the following stanar lemma on ranom currents. (2.2) Lemma 2.5 (Switching Lemma, [Aiz82, Lemma 3.2]). Let A Λ an u, v Λ {g}. Let F be a function from the set of currents on Λ to R. We have n =A {u,v} n 2 ={u,v} F (n + n 2 )w(n )w(n 2 ) = n =A where is the symmetric ifference between sets. F (n + n 2 )w(n )w(n 2 )I[u n +n 2 v], (2.3) Backbone representation for ranom currents. Fix a finite subset Λ of V. Choose an arbitrary orer of the oriente eges of the lattice. Consier a current n on Λ with n = {x, y}. Let ω(n) be the ege self-avoiing path from x to y passing only through eges e with n e o which is minimal for the lexicographical orer on paths inuce by the previous orering on oriente eges. Such an object is calle the backbone of the current configuration. For the backbone ω with enpoints ω = {x, y}, set ρ Λ (ω) = ρ Λ (β, h, ω) = n={x,y} w(n)i[ω(n) = ω]. n= w(n) The backbone representation has the following properties (see (4.2), (4.7) an (4.) of [AF86] for P, P2 an P3 respectively): P σ x σ y Λ,β,h = ω={x,y} ρ Λ (ω). P2 If the backbone ω is the concatenation of two backbones ω an ω 2 (this is enote by ω = ω ω 2 ), then ρ Λ (ω) = ρ Λ (ω )ρ Λ ω (ω 2 ), where ω is the set of bons whose state is etermine by the fact that ω is an amissible backbone (this inclues bons of ω together with some neighboring bons). P3 For the backbone ω not using any ege outsie T Λ, we have ρ Λ (ω) ρ T (ω). 2

13 2.4 Proof of Item In this section, we prove that for every β β c, σ 0 + β β 2 β c 2. (2.4) β 2 In orer to o so, we will base our analysis on the following lemma. Lemma 2.6. Let β > 0, h > 0 an Λ a finite subset of V. Then, β σ 0 2 Λ,β,h 2c(Λ, β, h) [ β inf ϕ β(s)( σ 0 2 Λ,β,h ) ɛ(λ, β, h)], (2.5) S 0 where an σ 0 Λ,β,h c(λ, β, h) = inf y Λ σ y Λ,β,h ɛ(λ, β, h) = J x,y ( σ 0 σ x Λ,β,h σ 0 Λ,β,h σ x Λ,β,h ). x Λ y Λ To conclue the proof, fix β, β 2 > β c. Integrating (2.5) between β an β 2 for Λ equal to the box Λ n of size n, an then letting Λ n go to infinity, implies that σ 0 2 β 2,h σ 0 2 β,h β 2 2 β β ( σ 0 2 β,h )β, where the inequality above follows from Fatou s lemma together with lim σ 0 Λn,β n i,h = σ 0 βi,h lim c(λ n, β, h) = n lim ɛ(λ n, β, h) = 0 n (by weak convergence), (see Remark 2.2 below), (see Remark 2.3 below). The proof of (2.4) follows easily by letting h ten to 0. Remark 2.2. To see that c(λ n, β, h) tens to, observe that Griffiths inequality (2.) implies that σ y Λn,β,h σ 0 Λ2n,β,h (we use the invariance uner translation an the fact that the translate of Λ 2n centere at y contains Λ n ). Therefore, for every n, we have σ 0 Λn,β,h c(λ n, β, h). (2.6) σ 0 Λ2n,β,h Together with the fact that σ 0 Λn,β,h tens to σ 0 β,h as n tens to infinity, (2.6) implies that c(λ n, β, h) tens to. Remark 2.3. To see that ɛ(λ n, β, h) tens to 0, first observe that the GHS inequality [GHS70] implies that σ 0 Λ,β,h is a concave function of h. We euce that σ 0 σ x Λ,β,h σ 0 Λ,β,h σ x Λ,β,h = x Λ h σ 0 Λ,β,h σ 0 Λ,β,h h h. 3

14 Applie to Λ = Λ n, this gives in particular that for each k, x Λ n k J x,y σ 0 σ x Λn,β,h σ 0 Λn,β,h σ x Λn,β,h y V Λ n h ( J 0,y ), y V Λ k which can be mae arbitrarily small (uniformly in n) by setting k large enough. Now, a secon use of the GHS inequality [GHS70] implies that σ 0 σ x Λn,β,h σ 0 Λn,β,h σ x Λn,β,h σ 0 Λn,β,h σ 0 Λn,β,h x Λ n Λ n k h σ 0 Λn,β,h σ 0 Λn k,β,h, h where Λnβ,h is the measure with inverse-temperature β, an magnetic fiel h x epening on x which is equal to h for x Λ n k an 0 in Λ n Λ n k. In the secon line, we use Griffiths inequality to show that σ 0 Λn k,β,h σ 0 Λn,β,h. For each fixe k, the term on the right converges to 0 as n tens to infinity by weak convergence. In orer to prove Lemma 2.6, we use a computation similar to one provie in [ABF87]. Proof of Lemma 2.6. Let β > 0, h > 0 an a finite subset Λ of V. Set Z = w(n). n= The erivative of σ 0 Λ,β,h is given by the following formula β σ 0 Λ,β,h = J x,y ( σ 0 σ x σ y Λ,β,h σ 0 Λ,β,h σ x σ y Λ,β,h ). {x,y} Λ Using (2.2) an the switching lemma, we obtain β σ 0 Λ,β,h = Z 2 {x,y} Λ J x,y n ={0,g} {x,y} w(n )w(n 2 )I[0 n +n 2 / g]. If n an n 2 are two currents such that n = {0, g} {x, y}, n 2 = an 0 an g are not connecte in n + n 2, then exactly one of these two cases hols: 0 n +n 2 x an y n +n 2 n +n 2 n +n 2 g, or 0 y an x g. Since the secon case is the same as the first one with x an y permute, we obtain the following expression, where δ x,y = n ={0,g} {x,y} β σ 0 Λ,β,h = Z 2 J x,y δ x,y, (2.7) x Λ y Λ w(n )w(n 2 )I[0 n +n 2 n +n n +n 2 2 x, y g, 0 / g] (see Fig. an notice the analogy with the event involve in Russo s formula, namely that the ege {x, y} is pivotal, in Bernoulli percolation). 4

15 g x y 0 Figure : A iagrammatic representation of δ x,y : the soli lines represent the backbones, an the otte line the bounary of the cluster of 0 in n + n 2. Given two currents n an n 2, an z {0, g}, efine S z to be the set of vertices in Λ {g} that are not connecte to z in n + n 2. Let us compute δ x,y by summing over the ifferent possible values for S 0 : δ x,y = S Λ {g} n ={0,g} {x,y} w(n )w(n 2 )I[S 0 = S, 0 n +n 2 n +n n +n 2 2 x, y g, 0 / g] = S Λ {g} s.t. y,g S an 0,x Λ S n ={0,g} {x,y} w(n )w(n 2 )I[S 0 = S, y n +n 2 g]. Since 0 an x are not connecte to y in n + n 2 (recall that y S), we euce that y must be connecte to g in n because of the constraints on sources. Thus, the inicator I[y n +n 2 g] equals for any currents n an n 2 satisfying S 0 = S. Therefore, δ x,y = S Λ {g} s.t. y,g S an 0,x Λ S n ={0,g} {x,y} w(n )w(n 2 )I[S 0 = S]. (2.8) Let us now focus on the following claim, which enables us to remove the sources y an g. Claim : Let S Λ containing y an g but neither x nor 0. We have n ={0,g} {x,y} w(n )w(n 2 )I[S 0 = S] (2.9) σ y Λ,β,h n ={0} {x} w(n )w(n 2 )I[S 0 = S, y n +n 2 g]. 5

16 Proof of Claim. Let Θ = n ={0,g} {x,y} w(n )w(n 2 )I[S 0 = S]. When S 0 = S, the two currents n an n 2 vanish on every {u, v} with u S an v S. Thus, for i =, 2, we can ecompose n i as n i = n S i + n Λ S i, where n A i enotes the current n A n i ({u, v}) if u, v A, i ({u, v}) = 0 otherwise. Note that n A i = A n i an w(n i ) = w(n Λ S i )w(n S i ). Since I[S 0 = S] oes not epen on n S, the ecomposition n = n S + nλ S gives Θ = Using (2.2), we fin Θ = n Λ S ={0} {x} n Λ S ={0} {x} w(n Λ S )w(n 2 )I[S 0 = S]( n S ={y,g} w(n S )). w(n Λ S )w(n 2 )I[S 0 = S] σ y S,β,h ( n S = w(n S )). Multiply the expression above by σ y Λ,β,h σ y S,β,h (which follows from (2.)), an then ecompose n 2 into n S 2 an nλ S 2 to fin σ y Λ,β,h Θ w(n Λ S )w(n 2 )I[S 0 = S] σ y 2 S,β,h ( w(n S )) n Λ S ={0} {x} n S = = n Λ S ={0} {x} n Λ S 2 = = n Λ S ={0} {x} n Λ S 2 = = n ={0} {x} {y,g} n 2 ={y,g} w(n Λ S )w(n Λ S 2 )I[S 0 = S] σ y 2 S,β,h w(n Λ S )w(n Λ S 2 )I[S 0 = S] ( w(n S )w(n S 2 )) n S = n S 2 = ( w(n S )w(n S 2 )) n S ={y,g} n S 2 ={y,g} w(n )w(n 2 )I[S 0 = S]. 6

17 The switching lemma (2.3) applie to F = I[S 0 = S] implies σ y Λ,β,h Θ n ={0} {x} w(n )w(n 2 )I[S 0 = S, y n +n 2 g]. Inserting (2.9) into (2.8) gives us δ x,y σ y Λ,β,h n ={0} {x} w(n )w(n 2 )I[y n n +n +n 2 2 g, 0 / g]. We now ecompose over the possible values of S g (recall that S g is the set of vertices not connecte to g): δ x,y = σ y Λ,β,h S Λ σ y Λ,β,h S Λ s.t. 0,x S an y Λ S n ={0} {x} n ={0} {x} w(n )w(n 2 )I[S g = S, y n n +n +n 2 2 g, 0 / g] w(n )w(n 2 )I[S g = S]. (2.0) In the secon line, we use the constraint on the sources, which implies that x is connecte to 0, an therefore, belong to S g. We now focus on a secon claim, which enables us to remove the sources 0 an x. Claim 2: Let S Λ containing 0 an x but neither y nor g. We have n ={0} {x} w(n )w(n 2 )I[S g = S] = n = w(n )w(n 2 ) σ 0 σ x S,0 I[S g = S]. Proof of Claim 2. For currents n an n 2 such that S g = S, n can be ecompose as n = n S + nλ {g} S as we i for S 0 = S in the previous claim. Using that w(n ) = w(n S )w(nλ {g} S ) together with the fact that I[S g = S] oes not epen on n S, we fin that n ={0} {x} w(n )w(n 2 )I[S g = S] = n Λ {g} S = = n Λ {g} S = = n = w(n Λ {g} S )w(n 2 )I[S g = S]( n S ={0} {x} w(n S )) w(n Λ {g} S )w(n 2 )I[S g = S]( n S = w(n S )) σ 0 σ x S,β,0 w(n )w(n 2 ) σ 0 σ x S,β,0 I[S g = S]. 7

18 In the thir line we use (2.2) an in the fourth line, we recombine n S with n Λ {g} S. Inequality (2.0) an Claim 2 imply that for any x, y Λ, δ x,y σ y Λ,β,h S Λ s.t. 0,x S an y Λ S n = By plugging the inequality above in (2.7), we fin β σ 0 2 Λ,β,h = 2 σ 0 Λ,β,h β σ 0 Λ,β,h 2 Z 2 S Λ S 0 2 Z 2 S Λ S 0 x S y Λ S 2c(Λ, β, h) Z 2 σ 0 Λ,β,h σ y Λ,β,h σ 0 Λ,β,h ( x S y Λ S S Λ S 0 n = w(n )w(n 2 ) σ 0 σ x S,β,0 I[S g = S]. w(n )w(n 2 )J x,y σ 0 σ x S,β,0 I[S g = S] σ y Λ,β,h J x,y σ 0 σ x S,β,0 )( ( x S y Λ S J x,y σ 0 σ x S,β,0 )( Using that J x,y β tanh(βj x,y) gives that We euce that J x,y σ 0 σ x S,β,0 β ϕ β(s) x S y Λ S β σ 0 2 Λ,β,h 2c(Λ, β, h)( β S Λ S 0 S Λ S 0 x S y V Λ J x,y σ 0 σ x S,β,0 Z 2 ϕ β (S) n = n = x S y V Λ Z 2 n = w(n )w(n 2 )I[S g = S]) w(n )w(n 2 )I[S g = S]). J x,y σ 0 σ x S,β,0. w(n )w(n 2 )I[S g = S] (A) n = w(n )w(n 2 )I[S g = S] ) (2.) (B) Taking the infimum over all the ϕ β (S) an then using (2.3) an (2.2) one more time, we obtain that (A) inf S 0 ϕ β(s) ( σ 0 2 Λ,β,h ). Now, summing on S after applying Claim 2 (backwar compare to the last use of Claim 2) gives that (B) = J x,y x Λ Z 2 y V Λ n ={0} {x} w(n )w(n 2 )( I[0 n +n 2 g]) = J x,y ( σ 0 σ x Λ,β,h σ 0 Λ,β,h σ x Λ,β,h ), x Λ y V Λ 8

19 where, in the secon line, we use (2.3) an (2.2) one last time. Plugging the expressions for (A) an (B) obtaine above in (2.) implies the claim. 2.5 Proof of Items 2 an 3 In this section, we show that Items 2 an 3 in Theorem 2. hol with β c in place of β c. We nee a replacement for the BK inequality use in the case of Bernoulli percolation. The relevant tool for the Ising moel will be a moifie version of Simon s inequality. The original inequality can be foun in [Sim80], see also [Lie80] for an improvement. (Those previous versions o not suffice for our application). Lemma 2.7 (Moifie Simon s inequality). Let S be a finite subset of V containing 0. For every z V S, σ 0 σ z + β tanh(βj x,y ) σ 0 σ x S,β,0 σ y σ z + β. x S y S Proof. Fix h 0 an Λ a finite subset of V containing S. We introuce the ghost vertex g as before. We consier the backbone representation of the Ising moel on Λ {g} efine in the previous section. Let ω = (v k ) 0 k K be a backbone from 0 to z (it may go through g). Since z S, one can efine the first k such that v k Λ S an set y = v k. Also set x to be the vertex of S visite last by the backbone before reaching y. The following occurs: ω goes from 0 to x staying in S {g}, then ω goes from x to y either in one step by using the ege {x, y} or in two steps by going through {x, g} an then {g, y}, finally ω goes from y to z in Λ {g}. Call ω the part of the walk ω from 0 to x, ω 2 the walk from x to y, an ω 3 the reminer of the walk ω. Using Property P of the backbone representation, we can write σ 0 σ z Λ,β,h = ρ Λ (ω). ω={0,z} Then, P2 applie with ω an ω 2 ω 3 an then with ω 2 an ω 3 implies that σ 0 σ z Λ,β,h is boune from above by x S y Λ S ω ={0,x} ρ Λ (ω )( ω 2 ={x,y} ρ Λ ω (ω 2 )( P an then Griffiths inequality (2.) imply that ω 3 ={y,z} ρ Λ ω ω 2 (ω 3 ))). ρ Λ ω ω 2 (ω 3 ) = σ y σ z Λ ω ω 2,β,h σ y σ z Λ,β,h. ω 3 ={y,z} Inserting this in the last isplaye equation gives σ 0 σ z Λ,β,h x S y Λ S ( ω ={0,x} ρ Λ (ω )( ω 2 ={x,y} ρ Λ ω (ω 2 ))) σ y σ z Λ,β,h. 9

20 Since ω 2 uses only vertices x, y an g, P3 an then P lea to which gives ρ Λ ω (ω 2 ) ρ {x,y} (ω 2 ) = σ x σ y {x,y},β,h ω 2 ={x,y} ω 2 ={x,y} σ 0 σ z Λ,β,h x S y Λ S ( ρ Λ (ω )) σ x σ y {x,y},β,h σ y σ z Λ,β,h. ω ={0,x} Finally, P3 can be use with the fact that ω S {g} to show that σ 0 σ z Λ,β,h x S y Λ S x S y Λ S ( ρ S (ω )) σ x σ y {x,y},β,h σ y σ z Λ,β,h ω ={0,x} σ 0 σ x S,β,h σ x σ y {x,y},β,h σ y σ z Λ,β,h (we use P in the secon line). Let Λ ten to V to obtain σ 0 σ z β,h Let now h ten to 0 to fin x S y V S σ 0 σ x S,β,h σ x σ y {x,y},β,h σ y σ z β,h. σ 0 σ z β,h an σ y σ z β,h ten to σ 0 σ z + β an σ yσ z + β respectively. σ 0 σ x S,β,h tens to σ 0 σ x S,β,0 (since S is finite). σ x σ y {x,y},β,h tens to σ x σ y {x,y},β,0 = tanh(βj x,y ). Using one last time that S is finite, we euce that σ 0 σ z + β x S y V S tanh(βj x,y ) σ 0 σ x S,β,0 σ y σ z + β. We are now in a position to conclue the proof. Let β < β c. Fix a finite set S such that ϕ β (S) <. Define, χ n (β) = sup { σ 0 σ z + β Λ V with Λ n}. z Λ By the same reasoning as for percolation, Lemma 2.7 shows that 0 σ z z Λ σ + β S + x S y V S tanh(βj x,y ) σ 0 σ x S,β,0 ( σ y σ z + β ). z Λ S Using the invariance uner translations an taking the supremum over sets Λ of volume n, we immeiately get that χ n (β) < S /[ ϕ β (S)] uniformly in n. Letting n ten to infinity gives the secon item. We finish by the proof of the thir item. Let R be the range of the (J x,y ) x,y V, an let L be such that S Λ L R. Lemma 2.7 implies that for any z with (0, z) n > L, σ 0 σ z + β x S y V S tanh(βj x,y ) σ 0 σ x S,β,0 σ y σ z + β ϕ β(s) max y Λ L σ y σ z + β. Note that (y, z) n L. If (y, z) L, we boun σ y σ z + β by, while if (y, z) > L, we apply the previous inequality to y an z instea of 0 an z. The proof follows by iterating n/l times this strategy. 20

21 2.6 Proof of Proposition 2.3 Let us introuce M(β, h) = σ 0 β,h. Recall that M(β, h) is ifferentiable in (β, h) away from the line h = 0. As in the case of percolation, the proof in [ABF87] invokes three inequalities (the pages below refer to the numbering in [ABF87]): the ifferential inequality (.2) page 348, M β ( J 0,y ) M M y V h, (2.2) the more ifficult ifferential inequality (.9) page 347, as well as (.3) page 348. Below, we combine Lemma 2.6 with (2.2) to conclue the proof without using (.9) or (.3) of [ABF87]. Since M(β, h) is ifferentiable for h > 0, we may pass to the limit Λ V in Lemma 2.6 to get M M β 2 β ( M 2 ) for h > 0 an β β c (once again we use that ϕ β (S) for any finite S 0 an for any β β c, see the comment before Proposition 2.2). Together with (2.2), we fin 2 β ( M 2 ) M M β ( J 0,y ) M 2 M y V h which immeiately implies that there exists a constant c > 0 such that for any h > 0, σ 0 βc,h = M(β c, h) ch /3. To conclue this article, let us recall the proof of (2.2) for completeness. Lemma 2.8 ((.2) page 348 of [ABF87]). On (0, ) (0, ), the function M satisfies the following ifferential inequality: Proof. Let β > 0, h > 0. We have M β ( J 0,y ) M M y V h. M β = J x,y ( σ 0 σ x σ y β,h σ 0 β,h σ x σ y β,h ). {x,y} The Griffith-Hurst-Sherman inequality [GHS70, (2.8)] gives σ 0 σ x σ y β,h σ 0 β,h σ x σ y β,h ( σ 0 σ x β,h σ 0 β,h σ x β,h ) σ y β,h + ( σ 0 σ y β,h σ 0 β,h σ y β,h ) σ x β,h. This implies that M β ( y V J 0,y ) M ( σ 0 σ x β,h σ 0 β,h σ x β,h ) = ( J 0,y ) M M x y V h. 2

22 Acknowlegments This work was supporte by a grant from the Swiss NSF an the NCCR SwissMap also fune by the Swiss NSF. We thank M. Aizenman an G. Grimmett for useful comments on this paper. We also thank D. Ioffe, A. Glazman an M. Lis for pointing out mistakes in previous versions of the manuscript. Finally, we thank the anonymous referees for numerous important comments an suggestions. References [AB87] M. Aizenman an D. J. Barsky. Sharpness of the phase transition in percolation moels. Comm. Math. Phys., 08(3): , 987. [ABF87] M. Aizenman, D. J. Barsky, an R. Fernánez. The phase transition in a general class of Ising-type moels is sharp. J. Statist. Phys., 47(3-4): , 987. [AF86] [Aiz82] [AN84] M. Aizenman an R. Fernánez. On the critical behavior of the magnetization in high-imensional Ising moels. J. Statist. Phys., 44(3-4): , 986. M. Aizenman. Geometric analysis of ϕ 4 fiels an Ising moels. I, II. Comm. Math. Phys., 86(): 48, 982. M. Aizenman an C. M. Newman. Tree graph inequalities an critical behavior in percolation moels. Journal of Statistical Physics, 36(- 2):07 43, 984. [AV08] T. Antunović an I. Veselić. Sharpness of the phase transition an exponential ecay of the subcritical cluster size for percolation on quasitransitive graphs. Journal of Statistical Physics, 30(5): , [BD2a] V. Beffara an H. Duminil-Copin. The self-ual point of the twoimensional ranom-cluster moel is critical for q. Probab. Theory Relate Fiels, 53(3-4):5 542, 202. [BD2b] V. Beffara an H. Duminil-Copin. Smirnov s fermionic observable away from criticality. Ann. Probab., 40(6): , 202. [BNP] I. Benjamini, A. Nachmias, an Y. Peres. Is the critical percolation probability local? Probab. Theory Relate Fiels, 49(-2):26 269, 20. [BR06] [CC87] Béla Bollobás an Oliver Rioran. A short proof of the Harris-Kesten theorem. Bull. Lonon Math. Soc., 38(3): , J. T. Chayes an L. Chayes. The mean fiel boun for the orer parameter of Bernoulli percolation. In Percolation theory an ergoic theory of infinite particle systems (Minneapolis, Minn., ), volume 8 of IMA Vol. Math. Appl., pages Springer, New York,

23 [DST5] H. Duminil-Copin, V. Sioravicius, an V. Tassion. Continuity of the phase transition for planar ranom-cluster an Potts moels with q 4. arxiv: , 205. [DT5] H. Duminil-Copin an V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation an the Ising moel. arxiv: , 205. [GHS70] Robert B. Griffiths, C. A. Hurst, an S. Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external fiel. J. Mathematical Phys., : , 970. [Gri67] [Gri99] [Gri06] R. B. Griffiths. Correlation in Ising ferromagnets I, II. J. Math. Phys., 8: , 967. G. Grimmett. Percolation, volume 32 of Grunlehren er Mathematischen Wissenschaften [Funamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, secon eition, 999. G. Grimmett. The ranom-cluster moel, volume 333 of Grunlehren er Mathematischen Wissenschaften [Funamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, [Ham57] J. M. Hammersley. Percolation processes: Lower bouns for the critical probability. Ann. Math. Statist., 28: , 957. [Har60] [Kes80] T. E. Harris. A lower boun for the critical probability in a certain percolation process. Proc. Cambrige Philos. Soc., 56:3 20, 960. H. Kesten. The critical probability of bon percolation on the square lattice equals 2. Comm. Math. Phys., 74():4 59, 980. [Lie80] E. H. Lieb. A refinement of Simon s correlation inequality. Comm. Math. Phys., 77(2):27 35, 980. [Men86] M. V. Menshikov. Coincience of critical points in percolation problems. Dokl. Aka. Nauk SSSR, 288(6):308 3, 986. [Ons44] L. Onsager. Crystal statistics. I. A two-imensional moel with an orer-isorer transition. Phys. Rev. (2), 65:7 49, 944. [Rus78] L. Russo. A note on percolation. Z. Wahrscheinlichkeitstheorie un Verw. Gebiete, 43():39 48, 978. [Sim80] B. Simon. Correlation inequalities an the ecay of correlations in ferromagnets. Comm. Math. Phys., 77(2): 26, 980. Département e Mathématiques Université e Genève Genève, Switzerlan hugo.uminil@unige.ch, vincent.tassion@unige.ch 23