Conformal Measure Ensembles and Planar Ising Magnetization: A Review
CCONFORMAL MEASURE ENSEMBLES ANDPLANAR ISING MAGNETIZATION: A REVIEW
FEDERICO CAMIA, JIANPING JIANG, AND CHARLES M. NEWMAN
Abstract.
We provide a review of results on the critical and near-critical scaling limitof the planar Ising magnetization field obtained in the past dozen years. The resultsare presented in the framework of coupled loop and measure ensembles, and some newproofs are provided. Synopsis
In [18] the first and third authors introduced the concept of Conformal Measure En-semble (CME) as the scaling limit of the collection of appropriately rescaled countingmeasures of critical FK-Ising clusters. They proposed to use a representation of the Isingmagnetization field in terms of such a CME to study its existence, uniqueness and con-formal properties in the critical scaling limit. Initial results and work in progress withChristophe Garban were presented by the first author at the Inhomogeneous RandomSystems 2010 conference (Institut Henri Poincar´e, Paris) and described in [6]. CMEsfor Bernoulli and FK-Ising percolation were first constructed in [7]. The results on thetwo-dimensional Ising model discussed or conjectured in [6] have now been fully provedand have appeared in various papers by a combination of different authors [7, 11–14].Those results, and more, were recently presented in a talk at the Inhomogeneous Ran-dom Systems 2020 conference (Institut Curie, Paris), which was the inspiration for thepresent paper, whose main goal is to review the results of [11–14] and present them inthe unifying CME framework. While the results presented in this paper are not new, insome cases their formulation is somewhat different than what has previously appeared inthe literature, and whenever we provide a detailed proof of a result, the proof is new.2.
Introduction and historical remarks
The Ising model was introduced by Lenz in 1920 [37] to describe ferromagnetism, andis nowadays one of the most studied models of statistical mechanics. The one-dimensionalversion of the model was studied by Ising in his Ph.D. thesis [28] and in his subsequentpaper [29], but it was not until Peierls’ and Onsager’s famous investigations of the two-dimensional version that the model gained popularity. In 1936 Peierls [44] proved thatthe two-dimensional model undergoes a phase transition; then in 1941 Kramers and Wan-nier [32] located the critical temperature of the model defined on the square lattice, andin 1944 Onsager [43] derived its free energy. Since then, the two-dimensional Ising modelhas played a special role in the theory of critical phenomena. Its phase transition hasbeen extensively studied by both physicists and mathematicians, becoming a prototypicalexample and a test case for developing ideas and techniques and for checking hypotheses.Ferromagnetism is one of the most interesting phenomena in solid state physics; itrefers to the tendency, observed in some metals such as iron and nickel, of the atomicspins to become spontaneuosly polarized in the same direction, generating a macroscopic
Mathematics Subject Classification.
Primary: 60K35, 82B20; Secondary: 82B27, 81T27, 81T40. a r X i v : . [ m a t h . P R ] N ov igure 1. The Ising model phase transition (courtesy of Wouter Kager).Blue (darker) and yellow (lighter) regions correspond to +1 and − Curie temperature . Above theCurie temperature the spins are oriented at random, producing no net magnetic field.Moreover, as the Curie temperature is approached from either side, the specific heat ofthe metal appears to diverge.The Ising model is a crude attempt to reproduce the behavior described above. Itsone-dimensional version fails to do so, as already realized by Ising [28, 29], but the two-dimensional version does exhibit a phase transition, as shown by Peierls [44] and subse-quently investigated by Kramers and Wannier [32], Onsager [43] and many others. Inthe most common version of the two-dimensional Ising model one associates a ± spin variable to each vertex of a square grid and then assigns to each spin configuration aprobability derived from a Gibbs distribution that favors the alignment of neighboringspins. The appeal of the two-dimensional version of the model stems from its simplic-ity and the fact that it yields to an exact treatment, which reveals a rich mathematicalstructure. Its analysis has provided important tests for various fundamental aspects ofthe theory of critical phenomena such as the scaling hypothesis , the emergence of scaleand conformal invariance at the critical point marking a phase transition, and Landau’smean-field theory. All these different aspects of the theory of critical phenomena finda natural interpretation in the renormalization group philosophy, which asserts that thecritical properties of a system do not depend on short-distance details but only on thenature of long-distance fluctuations, suggesting a coarse-graining procedure that removesthe short-distance features until one reaches the correlation length of the system, i.e., thecharacteristic length at which fluctuations become important and beyond which differentparts of the system become uncorrelated.The critical point is characterized by a diverging correlation length, as implied by Wu’scelebrated result [56] showing that the correlation length of the two-dimensional Isingmodel diverges as the critical point is approached and the two-point function betweenpositions x and y decays like | x − y | − / at criticality. This power-law behavior shouldbe contrasted with the exponential decay that, away from the critical point, implies theexistence of a finite correlation length (see Figure 1).The renormalization group coarse-graining procedure can take the form of a continuumscaling limit in which the mesh size of the grid on which the model is defined is sent tozero. At the critical point, where the correlation length diverges, Smirnov [52] provedthat certain observables of the two-dimensional Ising model have a well-defined scalinglimit which is conformally invariant. This groundbreaking and much celebrated result onfirms the emergence of conformal invariance and provides a link with conformal fieldtheory.The work of Chelkak, Hongler and Izyurov [19] and of Camia, Garban and Newman[11] can be seen as the culmination of this line of research: [19] proves the existenceand conformal invariance of the scaling limit of the n -point Ising correlation functionsand [11] shows that in the scaling limit the Ising magnetization converges to a conformallyinvariant Euclidean field.Conformal invariance is one of the most interesting features to emerge from the analysisof the scaling limit of critical models. Its emergence was predicted by Polyakov [45] andis discussed in [2, 3]. In two dimensions, conformal methods were applied extensivelyto Ising and Potts models, Brownian motion, the self-avoiding walk, percolation, anddiffusion limited aggregation.Moving away from criticality, one can modify the energy and hence the Gibbs dis-tribution of the Ising model with a linear function of the spin variables. This modelsthe presence of an external magnetic field that influences the alignment of the atomicspins. The model with an external field has never been solved exactly in dimension twoor higher. However, in two dimensions Zamolodchikov [57] proposed a solution of themodel directly in the scaling limit in the shape of a field theory containing eight massiveparticles whose masses are related to the exceptional Lie algebra E (see [5] for a “jour-nalistic” account of this relation). In [12] Camia, Garban and Newman showed that ameaningful scaling limit of this variant of the model can be obtained by scaling appropri-ately the external field with the lattice spacing in such a way that the correlation lengthshould remain bounded away from zero and infinity. In [13] the current authors showedthat the resulting field theory has a mass gap, providing a first step in the direction ofZamolodchikov’s theory.The main goal of this paper is to review the results of [11–14] and present them inthe unifying Conformal Measure Ensemble (CME) framework. The CME idea and itsusefulness in the study of scaling limits were first proposed in [18]. CMEs for Bernoulliand FK-Ising percolation were first constructed in [7]. The CME associated with FK-Isingpercolation plays a crucial role in the proof of exponential decay of [13].We point out that significant progress continues to be made in other directions thanthose mentioned above. Even the recent literature on the Ising model is too vast to besurveyed in this paper, but we refer the interested reader to [39] and references thereinfor a taste of recent results of a different flavor.3. The two-dimensional Ising model and its FK representation
We consider the standard Ising model on the square lattice a Z with (formal) Hamil-tonian (3.1) H = − (cid:88) { x,y } σ x σ y − H (cid:88) x σ x , where the first sum is over nearest-neighbor pairs in a Z , the spin variables σ x , σ y are( ± H is in R . For a bounded Λ ⊂ Z , the Gibbs distribution is given by Z Λ e − β H Λ , where H Λ is the Hamiltonian (3.1) with sums restricted to verticesin Λ, β ≥ inverse temperature , and the partition function Z Λ is the appropriatenormalization needed to obtain a probability distribution.The critical inverse temperature is β c = log (1 + √ β ≤ β c the modelhas a unique infinite-volume Gibbs distribution for any value of the external field H ,obtained as a weak limit of the Gibbs distribution for bounded Λ by letting Λ ↑ Z . For ny value of β ≤ β c and of H , expectation with respect to the unique infinite-volumeGibbs distribution will be denoted by (cid:104)·(cid:105) β,H . At the critical point , that is when β = β c and H = 0, expectation will be denoted by (cid:104)·(cid:105) c . By translation invariance, the two-pointcorrelation (cid:104) σ x σ y (cid:105) β,H is a function only of y − x . In particular, Wu [56] proved that (cid:104) σ x σ y (cid:105) c decays like | x − y | − / .We want to study the random field Φ a,H associated with the spins on the rescaledlattice a Z in the scaling limit a → a,H := Θ a (cid:88) x ∈ a Z σ x δ x , where δ x is a unit Dirac point measure at x and Θ a is an appropriate scale factor. Moreprecisely, for functions f of bounded support on R , we defineΦ a,H ( f ) := (cid:90) R f ( z )Φ a,H ( z ) dz := (cid:90) R f ( z ) (cid:2) Θ a (cid:88) x ∈ a Z σ x δ ( z − x ) (cid:3) dz (3.3) = Θ a (cid:88) x ∈ a Z f ( x ) σ x , (3.4)with scale factor Θ a proportional to(3.5) (cid:16) (cid:88) x,y ∈ [0 , ∩ a Z (cid:104) σ x σ y (cid:105) c (cid:17) − / . The rescaled block magnetization in a bounded domain D is M aD := Φ a,H ( I D ), where I denotes the indicator function. It is a rescaled sum of identically distributed dependent random variables. In the high-temperature regime, β < β c , and with zero externalfield, H = 0, the dependence is sufficiently weak for the rescaled block magnetizationto converge, as a →
0, to a mean-zero Gaussian random variable (see, e.g., [42] andreferences therein). In that case, the appropriate scaling factor Θ a is of order a , and thefield converges to Gaussian white noise as a → Zero external field.
The FK representation of the Ising model with zero externalfield, H = 0, is based on the q = 2 random cluster measure P F Kp (see [26] for more onthe random cluster model and its connection to the Ising model). A spin configurationdistributed according to the unique infinite-volume Gibbs distribution with H = 0 andinverse temperature β ≤ β c can be obtained in the following way. Take a random-cluster (FK) bond configuration on the square lattice distributed according to P F Kp with p = p ( β ) = 1 − e − β , and let {C ai } denote the corresponding collection of FK clusters,where a cluster is a maximal set of vertices of the square lattice connected via bonds ofthe FK bond configuration (see Figure 2). One may regard the index i as taking valuesin the natural numbers, but it’s better to think of it as a dummy countable index withoutany prescribed ordering, like one has for a Poisson point process. Let { η i } be ( ± σ x = η i for all x ∈ C ai ; then the collection { σ x } x ∈ a Z of spin variables is distributed according to the unique infinite-volume Gibbsdistribution with H = 0 and inverse temperature β . When β = β c , we will use thenotation P F Kc ≡ P F Kp ( β c ) , and E F Kc for expectation with respect to P F Kc .A useful property of the FK representation is that, when H = 0, the Ising two-pointfunction can be written as (cid:104) σ x σ y (cid:105) β, = P F Kp ( β ) ( x and y belong to the same FK cluster C ai ) . igure 2. Example of an FK bond configuration in a rectangular region(courtesy of Wouter Kager). Black dots represent sites of Z , black horizon-tal and vertical edges represent FK bonds. The FK clusters are highlightedby lighter (green) loops on the medial lattice.As an immediate consequence, we have that Θ − a is proportional to(3.6) (cid:88) x,y ∈ [0 , ∩ a Z (cid:104) σ x σ y (cid:105) c = (cid:88) x,y ∈ [0 , ∩ a Z E F Kc (cid:34)(cid:88) i x ∈C ai y ∈C ai (cid:35) = E F Kc (cid:34)(cid:88) i | ˆ C ai | (cid:35) , where ˆ C ai is the restriction of C ai to [0 , and | ˆ C ai | is the number of vertices of a Z in ˆ C ai .(Note that ˆ C ai need not be connected.) Using the FK representation and (3.3), we canwrite(3.7) Φ a, ( f ) dist. = (cid:88) i η i µ ai ( f ) , where µ ai := Θ a (cid:80) x ∈C ai δ ( z − x ) and the η i ’s are, as before, ( ± a was chosen so that the second moment of the rescaled block magnetization M a [0 , ,(3.8) (cid:68)(cid:2) Φ a ( I [0 , ) (cid:3) (cid:69) c = E F Kc (cid:104) (cid:88) i (cid:0) µ ai ( I [0 , ) (cid:1) (cid:105) = Θ a E F Kc (cid:104) (cid:88) i | ˆ C ai | (cid:105) , is bounded away from 0 and infinity. Wu’s celebrated result [56] on the decay of (cid:104) σ x σ y (cid:105) c implies that, for the two-dimensional Ising model at the critical point, we can take Θ a = a / so that µ ai := a / (cid:80) x ∈C ai δ ( z − x ) and M aD := a / (cid:80) x ∈ a Z ∩ D σ x .To each FK configuration we can associate a collection of loops on the medial latticeseparating the FK clusters from the dual clusters, where by dual clusters we mean max-imal connected subsets of dual bonds, and a dual bond is an edge of the dual graphcrossing perpendicularly a primal edge that contains no FK bond. See Figure 2 for anexample of an FK configuration with free boundary condition and the correspondingcollection of loops on the medial lattice. We call a (random) collection of loops associ-ated with an FK configuration in the way described above and shown in Fig. 2 a loopensemble . We denote by { γ ai } the collection of all loops associated with the FK clus-ters {C ai } . Each realization of { γ ai } can be seen as an element in a space of collections f loops with the Aizenman-Burchard metric [1]. (The latter is the induced Hausdorffmetric on collections of curves associated to the metric on curves given by the infimumover monotone reparametrizations of the supremum norm.) It follows from [1] and theRSW-type bounds of [21] (see Section 5.2 there) that, as a → { γ ai } has subsequentiallimits in distribution to random collections of loops in the Aizenman-Burchard metric.In the scaling limit, one gets collections of nested loops that can touch (themselves andeach other), but never cross.3.2. Non-zero external field.
Appropriate FK representations exist also for the Isingmodel on the square lattice (or, indeed, on any graph) with an external field H (cid:54) = 0. The“standard” one goes back to [23] and involves adding a vertex g , called the ghost vertex,connected to all vertices of the square lattice and carrying either a plus or a minus spin, σ g = ±
1, in accordance with the sign of the external field (see Section 4.3 of [23]). A spinconfiguration distributed according to the unique infinite-volume Gibbs distribution with H (cid:54) = 0 and inverse temperature β can be obtained by first taking a random-cluster (FK)bond configuration on the “augmented” square lattice obtained by adding the ghost spinas described above. In this case the distribution has two parameters: p = 1 − e − β for theedges between vertices of the square lattice, and p = 1 − e − βH for the edges connectingvertices of the square lattice to the ghost vertex. For each vertex x in a cluster connectedto g , one sets σ x = σ g . To clusters C ai not connected to g , one associates ( ± η i , and then sets σ x = η i for all x ∈ C ai . Then the collection { σ x } x ∈ a Z of spin variables is distributed according to the unique infinite-volume Gibbsdistribution with H (cid:54) = 0 and inverse temperature β .This representation is not useful in the scaling limit a ↓
0, as we will discuss later (seeRemark 3.3 below). For this reason we introduce a somewhat different FK representationexplained in detail in the next section. The latter is a direct consequence of the Edwards-Sokal coupling [22] and it appears implicitly in [20]. It was independently rediscoveredand developed in [13] and [14] where the authors observe how, contrary to the “standard”FK representation, it extends naturally to the near-critical scaling limit (see Remark 3.3below).3.3.
The Ising model in a bounded domain.
In this section we consider the Isingmodel on a finite graph D a = a Z ∩ D with free or plus boundary condition, where D isa bounded subset of R . The boundary of D a will be denoted by ∂D a := { x / ∈ D a : ∃ y ∈ D a with | x − y | = a } . In order to define the random cluster model associated with theIsing model (aka the FK-Ising model) on D a , we introduce the following sets of edges: E i = {{ x, y } : x, y ∈ D a , | x − y | = a } (3.9) E e = {{ x, y } : x ∈ D a , y ∈ ∂D a , | x − y | = a } (3.10) E = E i ∪ E e (3.11)We call the edges in E i and E e internal and external , respectively. The distribution ofthe collection of spins σ := { σ x } x ∈ D a for the Ising model on D a at inverse temperature β with external field H and free boundary condition is given by(3.12) P fβ,H ( σ ) := 1 Z fβ,H exp (cid:16) β (cid:88) { x,y }∈E i σ x σ y + βH (cid:88) x ∈ D a σ x (cid:17) , where(3.13) Z fβ,H := (cid:88) σ exp (cid:16) β (cid:88) { x,y }∈E i σ x σ y + βH (cid:88) x ∈ D a σ x (cid:17) s the partition function of the model. The distribution in the case of plus boundarycondition is given by(3.14) P + β,H ( σ ) := 1 Z + β,H exp (cid:16) β (cid:88) { x,y }∈E i σ x σ y + βH (cid:88) x ∈ D a σ x + β (cid:88) { x,y }∈E e : x ∈ D a σ x (cid:17) , where(3.15) Z + β,H := (cid:88) σ exp (cid:16) β (cid:88) { x,y }∈E i σ x σ y + βH (cid:88) x ∈ D a σ x + β (cid:88) { x,y }∈E e : x ∈ D a σ x (cid:17) . The configuration space of the random cluster model is { , } E . For each element ω ofthe configuration space and each edge e = { x, y } , we say that edge e is absent or closed if ω ( e ) = 0 and present or open if ω ( e ) = 1. We call ω a bond configuration . A cluster isa subset of D a ∪ ∂D a which is maximally connected using edges in E i ∪ E e . We denote by C aD the collection of clusters restricted to D a , that is(3.16) C aD = { ˜ C a ∩ D a : ˜ C a is a cluster in D a ∪ ∂D a } \ {∅} . To simplify the notation, in this section we write C i for an element of C aD instead of C ai as in other sections. This shouldn’t create any confusion since a is fixed, but the readershould be aware of the fact that in later sections C will be used to denote an element ofthe collection C D obtained from the scaling limit of C aD as a ↓ C ∈ C aD , we call C a boundary cluster if C is the restriction of a cluster ˜ C (in D a ∪ ∂D a ) which contains at least one element of ∂D a ; otherwise C is called an internal cluster .The boundary is treated differently depending on the choice of boundary condition, asfollows. Free b.c. : We set ω ( e ) = 0 for each e ∈ E e . In this case there are no boundary clusters. Wired b.c. : The vertices in ∂D a are identified and treated as a single vertex. Conse-quently, there is at most one boundary cluster. We denote by C b this boundary cluster ifit exists, and let |C b | denote the number of vertices in C b .We let ( ω ) = |{ e ∈ E : ω ( e ) = 1 }| denote the number of open edges in the FKconfiguration, ( ω ) = |{ e ∈ E : ω ( e ) = 0 }| denote the number of closed edges, and C ( ω )denote the number of internal clusters. With this notation, for any 0 ≤ p ≤
1, we definethe random cluster measure(3.17) P F Kp ( ω ) := 1 Z F Kp p ( ω ) (1 − p ) ( ω ) C ( ω ) , where Z F Kp := (cid:80) ω ∈{ , } E p ( ω ) (1 − p ) ( ω ) C ( ω ) is the partition function of the model. Therandom cluster measure corresponding to an Ising model on D a at inverse temperature β with zero external field ( H = 0) is given by (3.17) with the choice p = 1 − e − β . In thiscase we write(3.18) P F Kβ, ( ω ) := P F K − e − β ( ω ) = 1 Z F Kβ, (1 − e − β ) ( ω ) ( e − β ) ( ω ) C ( ω ) where Z F Kβ, := Z F K − e − β .The following statements are immediate consequences of the Edwards-Sokal couplingof FK percolation and the Ising model [22]. Free b.c. : A spin configuration on D a distributed according to P fβ, can be obtained by(1) sampling an FK configuration according to (3.18) with free boundary condition ,(2) sampling an independent, ( ± η i for each FKcluster C i ∈ C aD ,(3) for each cluster C i , setting σ x = η i for all x ∈ C i . lus b.c. : A spin configuration on D a distributed according to P + β, can be obtained by(1) sampling an FK configuration according to (3.18) with wired boundary condition ,(2) sampling an independent, ( ± η i for eachinternal cluster C i ,(3) for each internal cluster C i , setting σ x = η i for all x ∈ C i ,(4) setting σ x = 1 for all x belonging to the boundary cluster (if not empty).We note that coupled bond and spin configurations ( ω, σ ) generated by the Edwards-Sokal coupling are always compatible in the sense that, if x and y belong to the samebond cluster, then σ x = σ y . If a ω and σ are compatible, we write ω ∼ σ . This notationwill be used in Corollary 3.1 below and in its proof.In the rest of the section, we will use Z β,H to denote either Z fβ,H or Z + β,H . Moreover,letting ˜ M aD := (cid:80) x ∈ D a σ x denote the (“ bare ”) magnetization in D a and E β,H = (cid:104)·(cid:105) β,H theexpectation with respect to either P fβ,H or P + β,H , from (3.12) and (3.14) we have that, forany suitable function g ,(3.19) E β,H ( g ( σ )) = Z β, Z β,H E β, (cid:16) g ( σ ) e βH ˜ M aD (cid:17) . Taking the constant function g ( σ ) ≡
1, this gives(3.20) Z β,H = Z β, E β, (cid:16) e βH ˜ M aD (cid:17) = Z β, E F Kβ, ( e βH |C b | Π i (cid:54) = b cosh( βH |C i | )) , where |C b | = 0 if there is no boundary cluster, and the product is over all internal clustersin C aD .We will now present an FK representation for the Ising model with a non-zero externalfield that will be crucial when we discuss the near-critical scaling limit, in Section 5. Corollary 3.1 ( [13, 14] ) . Take β > and H > . For every ω ∈ { , } E with clusters {C i } , let (3.21) P F Kβ,H ( ω ) := e βH |C b | Π i (cid:54) = b cosh( βH |C i | ) E F Kβ, ( e βH |C b | Π i (cid:54) = b cosh( βH |C i | )) P F Kβ, ( ω ) . Furthermore, consider ( ± )-valued independent random variables { η i } such that η i = (cid:40) with probability e βH |C i | βH |C i | ) = tanh( βH |C i | ) + (cid:0) − tanh( βH |C i | ) (cid:1) − with probability e − βH |C i | βH |C i | ) = (cid:0) − tanh( βH |C i | ) (cid:1) (3.22) A spin configuration distributed according to P fβ,H can be obtained by sampling a con-figuration ω ∈ { , } E according to P F Kβ,H on D a with free boundary condition and, for eachcluster C i , setting σ x = η i for all x ∈ C i .A spin configuration distributed according to P + β,H can be obtained by sampling a con-figuration ω ∈ { , } E according to P F Kβ,H on D a with wired boundary condition, setting σ x = 1 for all x in the boundary cluster (if not empty) and, for each internal cluster C i ,setting σ x = η i for all x ∈ C i .The joint distribution of σ and ω is (3.23) P jointβ,H ( σ, ω ) := P F Kβ, ( ω ) E β, ( e βH ˜ M aD ) (cid:16) (cid:17) C ( ω ) e βH ˜ M aD ( σ ) I { ω ∼ σ } . Remark 3.2.
Corollary 3.1 is valid in any dimension and a similar result holds forarbitrary graphs. These results also extend from Ising models to Potts models — see theappendix of [14]. emark 3.3. As noted in Section 3.2, the “standard” FK representation for the Isingmodel with a non-zero external field involves adding a vertex g , called the ghost vertex,connected to all vertices of D a and carrying either a plus or a minus spin, σ g = ±
1, inaccordance with the sign of the external field (see Section 4.3 of [23]). This makes it un-suitable for taking scaling limits, in which the lattice spacing is sent to zero and individualvertices have no meaning. The representation discussed in Corollary 3.1, instead, reliesonly on “macroscopic” objects, namely the bond clusters. As we will see in Section 5,with the choice H = ha / , the appearance of the combination H |C i | in (3.21) and (3.22)and of H ˜ M aD = hM aD in (3.23) crucially allows us to make sense of this representation inthe near-critical scaling limit. Remark 3.4.
Equation (3.22) can be interpreted as saying that the spins in a bondcluster C i “follow” the sign of the external field H with probability tanh( βH |C i | ) andotherwise (i.e., with probability [1 − tanh( βH |C i | )]) take either +1 or − g carrying spin σ g = 1. In this case, however, the ghost vertex connects to FK clusters instead ofindividual vertices. Proof.
We begin by checking, with the help of the Edwards-Sokal coupling and of (3.20),that the marginal distribution induced by P jointβ,H on the bond configuration ω is P F Kβ,H : (cid:88) σ P jointβ,H ( σ, ω ) = P F Kβ, ( ω ) E β, ( e βH ˜ M aD ) (cid:16) (cid:17) C ( ω ) e βH |C b | (cid:88) { η i = ± i (cid:54) = b } e βH (cid:80) i η i |C i | (3.24) = P F Kβ, ( ω ) E β, ( e βH ˜ M aD ) e βH |C b | Π i : i (cid:54) = b (cid:16) (cid:88) η i = ± e βHη i |C i | (cid:17) = P F Kβ,H ( ω ) . (3.25)Next, using again (3.20) and the Edwards-Sokal coupling, we check that the marginaldistribution induced by P jointβ,H on the spin configuration σ is P Isingβ,H : P Isingβ,H ( σ ) = Z β, Z β,H P Isingβ, ( σ ) e βH ˜ M aD ( σ ) (3.26) = 1 E β, ( e βH ˜ M aD ) (cid:88) ω ∼ σ P F Kβ, ( ω ) (cid:16) (cid:17) C ( ω ) e βH ˜ M aD ( σ ) (3.27) = (cid:88) ω P jointβ,H ( σ, ω ) . (3.28)Finally, we check the distribution of the η i variables, given by (3.22). To do this, weintroduce the functions S i ( σ, ω ) defined on pairs of compatible configurations { ( σ, ω ) : ω ∼ σ } by letting S i ( σ, ω ) = σ x for any x ∈ C i , and note that ˜ M aD ( σ ) = (cid:80) i S i ( σ, ω ) |C i | .Using (3.21) and (3.23), and letting C ( x ) denote the FK cluster of x , we have that P jointβ,H ( σ x = ± | ω ) = P jointβ,H ( σ x = ± ω ) P F Kβ,H ( ω )(3.29) = 1Π i (cid:54) = b cosh( βH |C i | ) (cid:88) σ : σ x = ± Π i (cid:54) = b (cid:16) e βHS i ( σ,ω ) |C i | (cid:17) (3.30) = e ± βH |C ( x ) | βH |C ( x ) | ) . (3.31)This concludes the proof of the corollary. (cid:3) . The critical scaling limit
In this section we discuss the critical scaling limit of FK-Ising clusters and of the Isingmagnetization field. We restrict attention to bounded domains except where explicitlystated, as in Theorems 4.4 and 4.9 and Remarks 4.5 and 4.6.4.1.
Conformal loop and measure ensembles.
Theorem 1.1 of [31] shows that, atthe critical point, the (random) collection of loops { γ ai } coverges to a Conformal LoopEnsemble (CLE), namely CLE / , as a ↓
0. CLEs are random collections of closedcurves [15, 16, 48, 49, 54] which provide a useful tool to encode and analyze the scalinglimit geometry of, for instance, Bernoulli and FK percolation clusters, Ising spin clusters,and loops in the O ( n ) model. There is a one-parameter family of CLEs, CLE κ , indexedby a parameter κ ∈ [8 / , / < κ ≤
8, the loops of CLE κ locally “look like”SLE κ (see [46]). At the extremes, CLE almost surely consists of a single space-fillingloop, which is the scaling limit of the outer boundary of the free uniform spanning tree(see [34]), and CLE / almost surely contains no loops at all. When 8 / < κ <
8, thecollection of loops in a CLE κ is almost surely countably infinite. When κ = 6, it isequivalent to the random collection of loops described in [16], where it was shown to ariseas a scaling limit of the cluster boundaries of critical site percolation on the triangularlattice.The Schramm-Loewner Evolution (SLE) was introduced by Schramm [46] to describethe scaling limit of interfaces in critical models, where conformal invariance is believedto emerge in such a limit. The introduction of SLE, combined with the work of Smirnovon percolation [50] and of Lawler, Schramm and Werner on the loop erased random walkand the uniform spanning tree [34], spurred a flurry of activity and led to substantialprogress in the rigorous analysis of various critical models, including percolation (see,e.g., [8, 9, 15–17, 33, 53] for some of the early papers) and the Ising model (see [18, 51, 52]for a sample of the early work).Building on the convergence of { γ ai } to CLE / [31], Camia, Conijn and Kiss [7]provided the first construction of the CME for FK-Ising percolation. As we will showbelow, the FK-Ising CME can be used to construct a conformal field Φ which is anelement of an appropriate Sobolev space. Φ can be shown to correspond to the scalinglimit of the Ising magnetization at the critical point. We now introduce some notationand the relevant results from [31] and [7] before giving the construction of Φ .For any bounded domain D ⊂ R , recall from Section 3.3 that D a = a Z ∩ D denotesits a -approximation. Let L , L : [0 , → ¯ D be two loops in the closure ¯ D of D . Thedistance between L and L is defined by d loop ( L , L ) = inf sup t ∈ [0 , | L ( t ) − L ( t ) | , where the infimum is over all choices of parametrizations of L , L from the interval [0 , F and F , is defined by the Hausdorffmetric as follows: d LE ( F , F ) = inf { (cid:15) > ∀ L ∈ F , ∃ L ∈ F s.t. d loop ( L , L ) ≤ (cid:15) and vice versa } . The following theorem from [31] establishes the convergence of the critical FK-Isingloop ensemble on the medial lattice (see Figure 2 above and Section 1.2.2 of [31]) toCLE / . For a discrete domain D a = a Z ∩ D , we let Γ aD denote the FK-Ising loopensemble on the medial lattice. heorem 4.1 (Theorem 1.1 in [31]) . Consider critical FK-Ising percolation in a dis-crete domain D a with free or wired boundary condition. The FK-Ising loop ensemble Γ aD converges in distribution to CLE / in D in the topology of convergence defined by d LE . Remark 4.2.
A loop ensemble is a collection of loops on the medial lattice betweenFK clusters and dual clusters. Since the critical point is the self-dual point for FKpercolation, the critical FK loop ensembles for free and wired boundary conditions havethe same distribution.For any configuration ω in critical FK percolation on D a with free or wired bound-ary condition, let C aD denote the set of clusters of ω in D a . For C ai ∈ C aD , let µ ai := a / (cid:80) x ∈C ai δ x be the rescaled (by Θ a = a / ) counting measure of C ai , and let M aD = { µ ai } . For two collections, C and C , of subsets of ¯ D , the distance between C and C isdefined by(4.1) d cl ( C , C ) := inf { (cid:15) > ∀C ∈ C ∃C ∈ C s.t. d H ( C , C ) ≤ (cid:15) and vice versa } , where d H is the Hausdorff distance. Similarly, from two collections, S and S , ofmeasures on D , the distance between S and S is defined by(4.2) d meas ( S , S ) := inf { (cid:15) > ∀ µ ∈ S ∃ ν ∈ S s.t. d P ( µ, ν ) ≤ (cid:15) and vice versa } , where d P is the Prokhorov distance. The following theorem establishes convergence ofnormalized counting measures; the result follows directly from Theorems 11 and 13 (seealso Theorems 1 and 2 for simpler but slightly weaker versions), Theorem 14 and Lemma9 of [7]. Theorem 4.3 (Theorems 11, 13, 14 and Lemma 9 of [7]) . Let D be a bounded, simplyconnected domain. As a ↓ , ( C aD , M aD ) converge jointly in distribution to ( C D , M D ) where C D is a collection of subsets of ¯ D and M D is a collection of mutually orthogonalfinite measures such that for every C ∈ C D there is a µ C ∈ M D with supp( µ C ) = C . Thetopology of convergence is defined by d cl × d meas .Moreover, the joint law of (Γ aD , C aD , M aD ) converges in distribution to the joint law ofCLE / in D , C D and M D , with C D and M D measurable with respect to CLE / in D . We call the collection of measures M D in the previous theorem a CME / in D for itsrelation to CLE / in D .A full-plane version of CLE can be constructed either by taking an increasing sequenceof simply connected domains D n with ∪ ∞ n =1 D n = C , and for each n ∈ N letting Γ n be aCLE κ in D n , and then taking Γ to be the limit of Γ n as n → ∞ (see [41] for a detailedproof that the limit exists and does not depend on the sequence ( D n )), or equivalentlyby means of a branching SLE (see Section 2.3 of [27]). Theorem 3 of [7] shows thata full-plane CME / also exists. Combining these two results leads to the followingtheorem. Theorem 4.4.
Let P n denote the joint distribution of (Γ n , C n , M n ) where Γ n and M n area CLE / and its associated CME / in [ − n, n ] , respectively, and C n is the collection ofsupports of the measures in M n . There exists a probability measure P which is the full-plane limit of the probability measures P n in the sense that, for every bounded domain D ,the restriction P n | D of P n to D converges to the restriction P | D of P to D as n → ∞ . Remark 4.5.
In treating the full-plane versions of CLE and CME it is convenient toconsider the one-point (Alexandroff) compactification ˆ C of C , i.e., the Riemann sphere C := C ∪ {∞} , and to replace the Euclidean distance with(4.3) ∆( u, v ) := inf ϕ (cid:90) | ϕ (cid:48) ( s ) | | ϕ ( s ) | ds, where we take the infimum over all continuous differentiable paths ϕ ( s ) in C from u to v and |·| denotes the Euclidean norm. Doing this ensures that the sequence Γ n has a uniquelimit in distribution as n → ∞ by an application of Kolmogorov’s extension theorem (seethe proof of Theorem 3 of [7]). The joint convergence of (Γ n , C n , M n ) follows from thefact that C n and M n are measurable with respect to Γ n . Remark 4.6.
Theorem 4 of [7] shows that the full-plane collections of measures and theirsupports ( C , M ) that are the limits of ( C n , M n ) are conformally invariant/covariant. Thisexplains the name Conformal
Measure Ensemble (CME) for M . Similar considerationsapply to M D .4.2. The magnetization field.
Combining (3.7) with the results in the previous sub-section, it is tempting to try to define a continuum magnetization field as(4.4) Φ D ( f ) = (cid:88) C∈ C D η C µ C ( f ) , where µ C ∈ M D with supp( µ C ) = C and { η C } C∈ C D is a collection of independent randomvariables such that η C = 1 if C is a boundary cluster and η C is a ( ± D or functions f of bounded support, the sum above contains infinitely many terms, andthe scaling covariance of the µ C ’s suggests that the collection { µ C ( f ) } C∈ C D may in generalnot be absolutely summable.In order to make sense of (4.4), we introduce the ε -cutoff magnetization field(4.5) Φ D,ε ( f ) := (cid:88) C∈ C D :diam( C ) >ε η C µ C ( f ) , where the sum is now finite by Proposition 2.2 of [18] whenever D is bounded or f hasbounded support.Before stating the next result, which provides a precise meaning for (4.4), we need somepreliminaries. For a bounded domain D , let { u i } denote an eigenbasis of the negativeLaplacian (i.e., − ∆) on D with Dirichlet boundary condition with eigenvalues 0 ≤ λ ≤ λ ≤ . . . ↑ ∞ . The functions { u i } form an orthonormal basis of L ( D ) and of the classicalSobolev space H ( D ), and they satisfy (cid:107) u i (cid:107) H ( D ) = λ i . As a consequence, each f ∈ H ( D )has a unique orthogonal decomposition f = (cid:80) i a i u i with (cid:107) f (cid:107) H ( D ) = (cid:80) i | a i | λ i . Since C ∞ ⊂ H ( D ), the same holds for each f ∈ C ∞ ( D ). Moreover, if f ∈ C ∞ ( D ), then(4.6) (cid:88) i | a i | λ αi < ∞ , ∀ α > . To see why (4.6) holds, following [10], one can assume without loss of generalitythat α ≥ α and for every f ∈ C ∞ ( D ), one has that( − ∆) α f ∈ C ∞ ( D ), and consequently ( − ∆) α f = (cid:80) i (cid:104) ( − ∆) α f, u i (cid:105) L ( D ) u i , where the se-ries converges in L ( D ). Integration by parts yields moreover that (cid:104) ( − ∆) α f, u i (cid:105) L ( D ) = (cid:104) f, ( − ∆) α u i (cid:105) L ( D ) = λ αi (cid:104) f, u i (cid:105) L ( D ) , from which we deduce that(4.7) (cid:88) | a i | λ αi = (cid:104) ( − ∆) α f, f (cid:105) L ( D ) ≤ (cid:107) ( − ∆) α f (cid:107) L ( D ) · (cid:107) f (cid:107) L ( D ) < ∞ , as claimed. iven (4.6), one can define H α ( D ) to be the closure of C ∞ ( D ) with respect to the norm (cid:107) f (cid:107) H α ( D ) := (cid:80) i | a i | λ αi . The Sobolev space H − α ( D ) is then defined as the Hilbert dual of H α ( D ), that is, H − α ( D ) is the space of continuous linear functionals on H α ( D ), endowedwith the norm (cid:107) h (cid:107) H − α ( D ) := sup f : (cid:107) f (cid:107) H α D ) ≤ | h ( f ) | . One has that L ( D ) ⊂ H − α ( D ). Also,the action of h = (cid:80) i a i u i ∈ L ( D ) on f ∈ H α ( D ) is given by h ( f ) = (cid:82) D h ( z ) f ( z ) dz , andmoreover (cid:107) h (cid:107) H − α ( D ) = (cid:80) i λ − αi | a i | . Lemma 4.7.
Let D be a bounded, simply connected domain. For every α > , the cutofffield Φ D,ε converges as ε ↓ in second mean in the Sobolev space H − α ( D ) , in the sensethat there exists a H − α ( D ) -valued random field Φ D such that (4.8) lim ε ↓ E (cid:0) (cid:107) Φ D,ε − Φ D (cid:107) H − α ( D ) (cid:1) = 0 . Proof.
We are going to show that Φ D,ε is a Cauchy sequence in the Banach space of H − α ( D )-valued square-integrable random variables with norm (cid:113) E (cid:0) (cid:107) · (cid:107) H − α ( D ) (cid:1) (see, e.g.,[36]). The conclusion of the lemma then follows from the completeness of Banach spaces.We will use the fact that Φ D,ε is a.s. in L ( D ) for every ε > (cid:107) u i (cid:107) L ∞ ( D ) ≤ cλ i (see Theorem 1 of [25]), Weyl’s law [55] (which says that the number of eigenvalues λ i that are less than (cid:96) is proportional to (cid:96) with an error of o ( (cid:96) )), and the argument inthe proof of Proposition 2.1 of [18] (see also the argument leading to equation (10) in theproof of Theorem 7 of [7]).With these ingredients we have that, ∀ ε > ε (cid:48) > C, C (cid:48) < ∞ , E ( (cid:107) Φ D,ε − Φ D,ε (cid:48) (cid:107) H − α ( D ) ) = (cid:88) i λ αi E (cid:104)(cid:0) (Φ D,ε − Φ D,ε (cid:48) )( u i ) (cid:1) (cid:105) (4.9) ≤ (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ αi E (cid:104)(cid:16) (cid:88) C∈ C D : ε (cid:48) < diam( C ) ≤ ε η C µ C ( I D ) (cid:17) (cid:105) (4.10) ≤ (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ αi E (cid:104) (cid:88) C∈ C D :diam( C ) ≤ ε (cid:0) µ C ( I D ) (cid:1) (cid:105) (4.11) ≤ (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ αi lim a ↓ a / (cid:88) x,y ∈ D a : | x − y |≤ (cid:15) (cid:104) σ x σ y (cid:105) c (4.12) ≤ (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ αi Cε / (4.13) ≤ C (cid:48) ε / (cid:88) i λ α − / i . (4.14)For any α > / ε ↓
0, proving the claim. (cid:3)
Lemma 4.7 shows that it makes sense to define the continuum magnetization field in abounded domain D as an element of H − α ( D ):(4.15) Φ D = (cid:88) C∈ C D η C µ C where the right hand side is understood as the limit of Φ D,ε as ε ↓ he following theorem shows that the continuum magnetization field defined by (4.15)is the scaling limit of the lattice magnetization field. We note that Furlan and Mourrat [24]have proved that, for any (cid:15) >
0, the lattice magnetization field is tight in Besov spaces ofindex − / − (cid:15) but not − / (cid:15) . We remark that the scaling limit of the lattice energyfield is rather different — see Theorem 1.1 and Remarks 1.2 and 1.4 of [30]. Theorem 4.8.
Consider an Ising model at the critical point ( β = β c and H = 0 ) withfree or plus boundary condition on D a = D ∩ a Z where D is a bounded, simply connecteddomain of R . Then the lattice magnetization (4.16) Φ a, D := a / (cid:88) x ∈ D a σ x δ x converges in distribution to Φ D as a ↓ . The convergence is in any Sobolev space H − α ( D ) with α > in the topology induced by the norm (cid:107) · (cid:107) H − α ( D ) .Proof. To simplify the notation, in this proof we write Φ aD for Φ a, D . We first show thatΦ aD has subsequential limits in distribution in the topology induced by (cid:107) · (cid:107) H − α ( D ) , for any α > /
2. We will make use of Rellich’s theorem, which implies that H − α ( D ) is compactlyembedded in H − α ( D ) for any α < α and thus, in particular, that the closure of a ballof finite radius in H − α ( D ) is compact in H − α ( D ).Given α > /
2, let (cid:15) = α − / > α (cid:48) = 3 / (cid:15)/ < α . A straightforwardcalculation, using some of the ingredients of the proof of Lemma 4.7, shows thatlim sup a ↓ E (cid:16) (cid:107) Φ aD (cid:107) H − α (cid:48) ( D ) (cid:17) = lim sup a ↓ (cid:88) i λ α (cid:48) i E (cid:16)(cid:0) Φ aD ( u i ) (cid:1) (cid:17) (4.17) ≤ (cid:16) (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ α (cid:48) i (cid:17) lim sup a ↓ E (cid:16)(cid:0) Φ aD ( I D ) (cid:1) (cid:17) (4.18) ≤ c (cid:16) (cid:88) i λ (cid:15)/ i (cid:17) lim sup a ↓ E (cid:16)(cid:0) Φ aD ( I D ) (cid:1) (cid:17) , (4.19)where the last inequality follows from the bound in Theorem 1 of [25] and the lastexpression is finite because of Weyl’s law [55] and the boundedness of the second momentof the rescaled magnetization (see the discussion leading to (3.8)). This calculation,combined with Chebyshev’s inequality and Rellich’s theorem, implies that Φ aD is tight in H − α ( D ) for any α > / D defined by (4.15). Let ˜Φ D denote any such subsequential limit obtained from a convergingsequence Φ a k D . For any ε >
0, using the Edwards-Sokal coupling, we can write(4.20) Φ a k D = Φ a k D,ε + (cid:88) i :diam( C i ) ≤ ε η i µ a k i , where Φ a k D,ε := (cid:80) i :diam( C ai ) >ε η i µ a k i . Since E (cid:16) (cid:107) Φ aD,ε (cid:107) H − α (cid:48) ( D ) (cid:17) ≤ E (cid:16) (cid:107) Φ aD (cid:107) H − α (cid:48) ( D ) (cid:17) , the ar-gument above shows that Φ aD,ε is tight in H − α ( D ) as a ↓ α > /
2. Proposition2.2 of [18] implies that, for any f ∈ C ∞ ( D ), the number of elements in the sum definingΦ aD,ε ( f ) remains finite as a ↓
0; hence for every f ∈ C ∞ ( D ), by Theorem 4.3, Φ aD,ε ( f )converges in distribution to Φ D,ε ( f ) as a ↓
0. This, combined with the tightness of Φ aD,ε and with Lemma A.4 of [10], shows that, as k → ∞ , Φ a k D,ε converges to Φ D,ε in distribu-tion in the topology induced by (cid:107) · (cid:107) H − α ( D ) . Therefore, (cid:80) i :diam( C i ) ≤ ε η i µ a k i also convergesin distribution in the topology induced by (cid:107) · (cid:107) H − α ( D ) to some ˜ X ε ∈ H − α ( D ), and Φ a k D onverges to ˜Φ D = Φ D,ε + ˜ X ε . Moreover, a calculation analogous to that carried out inthe proof of Lemma 4.7 shows that E ( (cid:107) ˜ X ε (cid:107) H − α ( D ) ) ≤ lim sup a k ↓ E β c , (cid:16)(cid:13)(cid:13)(cid:13) (cid:88) i :diam( C i ) ≤ ε η i µ a k i ( I D ) (cid:13)(cid:13)(cid:13) H − α ( D ) (cid:17) (4.21) ≤ (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ αi lim sup a k ↓ E F Kβ c , (cid:104) (cid:88) i :diam( C i ) ≤ ε (cid:0) µ a k i ( I D ) (cid:1) (cid:105) (4.22) ≤ C (cid:48) ε / (cid:88) i λ α − / i → ε → . (4.23)We note that the processes Φ D , Φ D,ε , ˜Φ D , and ˜ X ε are all measurable with respect toCLE / in D and have a joint distribution on the space of conformal loop and measureensembles. Hence, using the triangle inequality and Jensen’s inequality, we have that E ( (cid:107) Φ D − ˜Φ D (cid:107) H − α ( D ) ) ≤ E ( (cid:107) Φ D − Φ D,ε (cid:107) H − α ( D ) ) + E ( (cid:107) ˜ X ε (cid:107) H − α ( D ) )(4.24) ≤ E ( (cid:107) Φ D − Φ D,ε (cid:107) H − α ( D ) ) + (cid:16) C (cid:48) ε / (cid:88) i λ α − / i (cid:17) / . (4.25)Markov’s inequality now implies that, for any δ > P (cid:0) (cid:107) Φ D − ˜Φ D (cid:107) H − α ( D ) > δ (cid:1) can bemade arbitrarily small using the bound (4.25) together with (4.23) and Lemma 4.7, andtaking ε small. This concludes the proof of uniqueness of subsequential limits and of thetheorem. (cid:3) Theorem 4.8 and equation (4.4) provide a sort of continuum FK representation forthe magnetization field. A related result was recently proved by Miller, Sheffield andWerner [40]. Theorem 7.5 of [40] shows that forming clusters of CLE / loops by a per-colation process with parameter p = 1 / , the Conformal Loop Ensemblewith parameter 3. CLE describes the full scaling limit of Ising spin-cluster boundaries [4]while CLE / , as already mentioned, describes the full scaling limit of FK-Ising clusterboundaries [31]. We note that, although the magnetization can obviously be expressedusing Ising spin clusters, as a sum of their signed areas, such a representation does notappear to be useful in the scaling limit because the area measures of spin clusters scalewith exponent 187 /
96 by [47] while the magnetization scales with exponent 15 /
8. Theusefulness of the representation in terms of FK clusters is due to the fact that both theFK clusters and the magnetization need to be multiplied by the same scale factor inthe scaling limit to obtain meaningful nontrivial limits. That is not the case for themagnetization and the spin clusters.To conclude this section we note that, using Theorem 4.4, it is possible to consider afull-plane version of the magnetization, which one can denote(4.26) Φ = (cid:88) C∈ C η C µ C where C denotes the full-plane collection of continuum clusters. To be more precise,consider the magnetization field Φ n := Φ − n,n ] . Given f ∈ C ∞ , let A f,k denote theevent that no continuum cluster intersects both the support of f and the complement of[ − k, k ] . Using Theorem 4.4, for any k < n , we can write the distribution function of n ( f ) as P n (Φ n ( f ) ≤ x )= P n (Φ n ( f ) ≤ x and A f,k ) + P n (Φ n ( f ) ≤ x and A cf,k )(4.27) = P n (cid:12)(cid:12) [ − k,k ] (cid:16) (cid:88) C∈ C : C∩ supp( f ) (cid:54) = ∅ η C µ C ( f ) ≤ x and A f,k (cid:17) (4.28) + P n (Φ n ( f ) ≤ x and A cf,k ) . The fact that there is no infinite cluster in critical FK-Ising percolation implies that onecan make the term P n (Φ n ( f ) ≤ x and A cf,k ) arbitrarily small by taking n and k sufficientlylarge. Theorem 4.4 implies that P n (cid:12)(cid:12) [ − k,k ] (cid:16) (cid:80) C∈ C : C∩ supp( f ) (cid:54) = ∅ η C µ C ( f ) ≤ x and A f,k (cid:17) con-verges to P (cid:12)(cid:12) [ − k,k ] (cid:16) (cid:80) C∈ C : C∩ supp( f ) (cid:54) = ∅ η C µ C ( f ) ≤ x and A f,k (cid:17) as n ↑ ∞ . This last proba-bility converges to P (cid:16) (cid:80) C∈ C : C∩ supp( f ) (cid:54) = ∅ η C µ C ( f ) ≤ x (cid:17) =: P (Φ ( f ) ≤ x ) as k ↑ ∞ . Thisshows that lim n ↑∞ P n (Φ n ( f ) ≤ x ) = P (Φ ( f ) ≤ x ).Theorem 4 of [7] implies that the full-plane magnetization (4.26) is conformally covari-ant. In particular, one has that(4.29) Φ( I [ − αL,αL ] ) dist = α / Φ( I [ − L,L ] )where dist = denotes equality in distribution. An equivalent way to express the conformalcovariance of the magnetization field is presented in the next theorem, where, with aslight abuse of notation, we write Φ ( x ) even though Φ is not defined pointwise. Theorem 4.9 (Theorem 4.1 of [13]) . For any λ > , the field Φ λ ( x ) = Φ ( λx ) given by Φ λ ( f ) = (cid:90) R Φ ( λx ) f ( x ) dx (4.30) = (cid:90) R Φ ( y ) f ( λ − y ) λ − dy = λ − Φ ( f λ − ) , (4.31) with f λ − ( x ) = f ( λ − x ) is equal in distribution to λ − / Φ ( x ) . Theorem 4.9 is a special case of Theorem 1.8 of [11], which shows that the distributionof the magnetization field transforms covariantly under any conformal map between anytwo simply connected domains. For more details, we refer the reader to Theorem 1.8 andCorollary 1.9 of [11], as well as Section 4.2 of [13].5.
The near-critical scaling limit
CME / describes the continuum scaling limit of the rescaled counting measures asso-ciated with critical FK-Ising clusters. As we have seen in the previous section, CME / provides, via equation (4.4), a sort of continuum FK representation for the critical magne-tization field. Surprisingly, CME / , also plays a crucial role in the proof of exponentialdecay for the near-critical magnetization field in [13]. The tool that allows the use ofCME / in the analysis of the near-critical scaling limit is the coupling presented inCorollary 3.1.Let P denote the law of CME / in a bounded, simply connected domain D , and E denote expectation with respect to P . Equation (3.21), combined with Theorem 4.3 andwith Corollary 3.8 of [11], shows that in the near-critical regime, β = β c and H = ha / , s a ↓
0, the distribution of the collection of FK clusters in D has a limit P h defined bythe Radon-Nikodym derivative dP h dP = exp( β c hµ C b ( I D ))Π C∈ C D : C(cid:54) = C b cosh( β c hµ C ( I D )) E (exp( β c hµ C b ( I D ))Π C∈ C D : C(cid:54) = C b cosh( β c hµ C ( I D )))(5.1) = exp( β c hµ C b ( I D ))Π C∈ C D : C(cid:54) = C b cosh( β c hµ C ( I D )) E ( e β c hM D ) , (5.2)where M D := Φ D ( I D ) and we have used a near-critical scaling limit version of (3.20) inthe last equality. We let { µ h C } C∈ C D denote a collection of measures distributed accordingto P h ; they are the scaling limit of the rescaled counting measures of the FK clusters in D with free or wired boundary conditions in the near-critical regime.In analogy with (4.15), we would like to define a near-critical magnetization field(5.3) Φ hD = (cid:88) C∈ C D η h C µ h C , where { η h C } C∈ C D is a collection of independent random variables such that η C = 1 if C isa boundary cluster and otherwise η C has distribution given by η h C = e βchµh C ( I D ) β c hµ h C ( I D )) = tanh( β c hµ h C ( I D )) + (cid:0) − tanh( β c hµ h C ( I D )) (cid:1) − e − βchµh C ( I D ) β c hµ h C ( I D )) = (cid:0) − tanh( β c hµ h C ( I D )) (cid:1) (5.4)To make sense of (5.3) we follow the same strategy we used in the case of the criticalmagnetization (4.15): we define a near-critical ε -cutoff field(5.5) Φ hD,ε := (cid:88) C∈ C D :diam( C ) >ε η h C µ h C and show that it has a limit as ε ↓
0. With these definitions, we can write a usefulnear-critical scaling limit version of (3.19) and (3.20), namely,(5.6) E (cid:0) g (cid:0) Φ hD,ε ( f ) (cid:1)(cid:1) = E (cid:0) g (cid:0) Φ D,ε ( f ) (cid:1) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) for any suitable functions f and g . Lemma 5.1.
Let D be a bounded, simply connected domain. For every α > , the cutofffield Φ hD,ε converges as ε ↓ in mean in the Sobolev space H − α ( D ) , in the sense that thereexists a H − α ( D ) -valued random field Φ hD such that (5.7) lim ε ↓ E (cid:0) (cid:107) Φ hD,ε − Φ hD (cid:107) H − α ( D ) (cid:1) = 0 . Proof.
The proof is analogous to that of Lemma 4.7, so we point out the only difference,which consists in replacing the L bound (4.14) with the following L bound, which uses(5.6), the Cauchy-Schwarz inequality, (4.14) and the fact that E (cid:0) e β c rM D (cid:1) is finite for any r ≥ roposition 3.5 in [11]): E ( (cid:107) Φ hD,ε − Φ hD,ε (cid:48) (cid:107) H − α ( D ) ) = E (cid:16) (cid:107) Φ D,ε − Φ D,ε (cid:48) (cid:107) H − α ( D ) e β c hM D (cid:17) E (cid:0) e β c hM D (cid:1) (5.8) ≤ (cid:114) E (cid:16) (cid:107) Φ D,ε − Φ D,ε (cid:48) (cid:107) H − α ( D ) (cid:17) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) (5.9) ≤ (cid:113) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) (cid:16) C (cid:48) (cid:88) i λ α − / i (cid:17) / ε / . (5.10)The rest of the proof is the same as the proof of Lemma 4.7. (cid:3) Lemma 5.1 shows that it makes sense to define the near-critical continuum magnetiza-tion field in a bounded domain D as an element of H − α ( D ):(5.11) Φ hD = (cid:88) C∈ C D η h C µ h C where the right hand side is understood as the limit of Φ hD,ε as ε ↓
0. Next, we will showthat (5.11) is the near-critical scaling limit of the lattice magnetization field.
Theorem 5.2.
Consider an Ising model in the near-critical regime ( β = β c and H = ha / for some h > ) with free or plus boundary condition on D a = D ∩ a Z where D is a bounded, simply connected domain of R . Then the lattice magnetization (5.12) Φ a,HD := a / (cid:88) x ∈ D a σ x δ x converges in distribution to Φ hD as a ↓ . The convergence is in any Sobolev space H − α ( D ) with α > in the topology induced by the norm (cid:107) · (cid:107) H − α ( D ) .Proof. The proof follows the same strategy as that of Theorem 4.8, so we only pointout the differences. The first difference is that, in proving the tightness of the lattice agnetization, we need to replace the bounds leading to (4.19) withlim sup a ↓ E β c ,H (cid:16) (cid:107) Φ a,HD (cid:107) H − α (cid:48) ( D ) (cid:17) = lim sup a ↓ E β c , (cid:16) (cid:107) Φ a, D (cid:107) H − α (cid:48) ( D ) e β c HM aD (cid:17) E β c , (cid:0) e β c HM aD (cid:1) (5.13) ≤ E (cid:0) e β c hM D (cid:1) lim sup a ↓ (cid:114) E β c , (cid:16) (cid:107) Φ a, D (cid:107) H − α (cid:48) ( D ) (cid:17) E β c , (cid:0) e β c HM aD (cid:1) (5.14) ≤ (cid:113) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) lim sup a ↓ (cid:115) E β c , (cid:104)(cid:16) (cid:88) i λ α (cid:48) i (cid:0) Φ a, D ( u i ) (cid:1) (cid:17) (cid:105) (5.15) ≤ (cid:113) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) lim sup a ↓ (cid:118)(cid:117)(cid:117)(cid:116)(cid:16) (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ α (cid:48) i (cid:17) E β c , (cid:104)(cid:0) M aD (cid:1) (cid:105) (5.16) ≤ (cid:113) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) (cid:16) (cid:88) i (cid:107) u i (cid:107) L ∞ ( D ) λ α (cid:48) i (cid:17) lim sup a ↓ (cid:114) E β c , (cid:104)(cid:0) M aD (cid:1) (cid:105) (5.17) ≤ (cid:113) E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) c (cid:16) (cid:88) i λ (cid:15)/ i (cid:17) lim sup a ↓ (cid:114) E β c , (cid:104)(cid:0) M aD (cid:1) (cid:105) , (5.18)where we have used (3.19), the Cauchy-Schwarz inequality and (4.19). The last expressionis finite because of Weyl’s law [55] and the boundedness of the fourth moment of therenormalized magnetization (see, e.g., Corollary 3.8 of [11]).In order to show that all subsequential limits coincide with the field Φ hD defined by(5.11), we let ˜Φ hD denote any such subsequential limit obtained from a converging sequenceΦ a k ,HD . We then use the coupling in Corollary 3.1 to write(5.19) Φ a k ,HD = Φ a k ,HD,ε + (cid:88) i :diam( C aki ) ≤ ε η i µ a k i , where Φ a k ,HD,ε := (cid:80) i :diam( C aki ) >ε η i µ a k i .The argument then proceeds as in the proof of Theorem 4.8, but with the boundsleading to (4.23) replaced by E β c ,H ( (cid:107) ˜ X ε (cid:107) H − α ( D ) ) ≤ lim sup a k ↓ E β c ,H (cid:16)(cid:13)(cid:13)(cid:13) (cid:88) i :diam( C aki ) ≤ ε η i µ a k i ( I D ) (cid:13)(cid:13)(cid:13) H − α ( D ) (cid:17) (5.20) = lim sup a k ↓ E β c , (cid:16)(cid:13)(cid:13)(cid:13) (cid:80) i :diam( C aki ) ≤ ε η i µ a k i ( I D ) (cid:13)(cid:13)(cid:13) H − α ( D ) e β c HM aD (cid:17) E β c , (cid:0) e β c HM aD (cid:1) (5.21) ≤ E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) lim sup a k ↓ E β c , (cid:16)(cid:13)(cid:13)(cid:13) (cid:88) i :diam( C aki ) ≤ ε η i µ a k i ( I D ) (cid:13)(cid:13)(cid:13) H − α ( D ) (cid:17) (5.22) ≤ E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) C (cid:48) ε / (cid:88) i λ α − / i → ε → , (5.23) here we have used (3.19) and the Cauchy-Schwarz inequality, as before, and (4.22) and(4.23) in the last line. We then have that E ( (cid:107) Φ hD − ˜Φ hD (cid:107) H − α ( D ) ) ≤ E ( (cid:107) Φ hD − Φ hD,ε (cid:107) H − α ( D ) ) + E ( (cid:107) ˜ X ε (cid:107) H − α ( D ) )(5.24) ≤ E ( (cid:107) Φ hD − Φ hD,ε (cid:107) H − α ( D ) ) + E (cid:0) e β c hM D (cid:1) E (cid:0) e β c hM D (cid:1) C (cid:48) ε / (cid:88) i λ α − / i . (5.25)Markov’s inequality now implies that, for any δ > P (cid:0) (cid:107) Φ hD − ˜Φ hD (cid:107) H − α ( D ) > δ (cid:1) can bemade arbitrarily small using the bound (5.25) together with (5.23) and Lemma 5.1, andtaking ε small. This concludes the proof of uniqueness of subsequential limits and of thetheorem. (cid:3) As in the zero external field case (see (4.26) and the discussion at the end of Section 4.2),one can consider a full-plane version of the magnetization field:(5.26) Φ h = (cid:88) C∈ C η h C µ h C . The main ingredients in the construction of Φ h are RSW, FKG and stochastic domination.Using those ingredients, for any n ∈ N and ε >
0, one can find an m ∈ N sufficientlylarge and a coupling between an Ising configuration on a Z and an Ising configuration in[ − m, m ] ∩ a Z with plus boundary condition such that the two configurations coincideinside [ − n, n ] with probability at least 1 − ε . A discussion of this construction can befound in the proof of Lemma 4.1 of [12].Contrary to the critical magnetization field Φ , the near-critical field Φ h is not confor-mally covariant and the considerations at the very end of Section 4.2 do not apply to it.Indeed, Φ h exhibits exponential decay of correlations, or a mass gap in the language offield theory. This important result, proved in [13], is too complex to discuss in detail andprovide a complete proof here. However, below we comment briefly on it and its proof.Let (cid:104) σ x ; σ y (cid:105) β,H denote the Ising truncated two-point function, i.e.,(5.27) (cid:104) σ x ; σ y (cid:105) β,H := (cid:104) σ x σ y (cid:105) β,H − (cid:104) σ x (cid:105) β,H (cid:104) σ y (cid:105) β,H . The exponential decay of correlations in the near-critical field Φ h is proved in [13] as aconsequence of the following result. Theorem 5.3 (Theorem 1.1 of [13] and Theorem 1 of [14]) . Consider the Ising modelon a Z at inverse temperature β = β c and with external field H = ha / . There exists B , B , C , C ∈ (0 , ∞ ) such that, for any a ∈ (0 , and h ∈ (0 , a − / ] , (5.28) C a / h / e − B h / | x − y | ≤ (cid:104) σ x ; σ y (cid:105) β,H ≤ C a / | x − y | − / e − B h / | x − y | for any x, y ∈ a Z . The exponential decay of the Ising truncated two-point function for
H >
Theorem 5.4 (Theorem 1.4 of [13]) . For any f, g ∈ C ∞ , we have (5.29) | Cov(Φ h ( f ) , Φ h ( g )) | ≤ C (cid:90) (cid:90) R × R | f ( x ) || g ( y ) || x − y | − / e − B h / | x − y | dxdy, where B and C are as in Theorem 5.3. lthough the near-critical magnetization field is not conformally covariant like thecritical one, it still possesses interesting scaling properties, as shown by the next result. Theorem 5.5 (Theorem 4.2 of [13]) . For any λ > and h > , the field λ / Φ h ( λx ) isequal in distribution to Φ λ / h ( x ) . The following observation may be useful to interpret Theorem 5.5. As discussed at theend of Section 4.2, in the zero-field case, Φ ( λx ) is equal in distribution to λ − / Φ ( x ) inthe sense that, with the change of variables z = λx , (cid:90) Φ ( z ) f ( z ) dz dist = (cid:90) λ − / Φ ( x ) f ( λx ) λ dx dist = λ / (cid:90) Φ ( x ) f ( λx ) dx for any f ∈ C ∞ ( R ), where the equalities are in distribution. In the non-zero-field case,provided that ˜ h = λ − / h , using Theorem 5.5 one obtains an analogous relation asfollows: (cid:90) Φ ˜ h ( z ) f ( z ) dz = (cid:90) Φ λ − / h ( λx ) f ( λx ) λ dx = λ / (cid:90) λ / Φ λ − / h ( λx ) f ( λx ) dx = λ / (cid:90) Φ h ( x ) f ( λx ) dx. We now consider the field Φ hD in a simply connected domain not equal to C and aconformal map φ : D → ˜ D with inverse ψ = φ − : ˜ D → D . The pushforward by φ of Φ D to a generalized field on ˜ D is described explicitly in Theorem 1.8 of [11]. Thegeneralization to Φ h , implicit in [12], is stated explicitly in the next theorem, takenfrom [13], where we introduce the non-constant magnetic fields h ( z ) and ˜ h ( x ). Theorem 5.6 (Theorem 4.3 of [13]) . The field Φ hD,ψ ( x ) := Φ hD ( ψ ( x )) on ˜ D is equal indistribution to the field | ψ (cid:48) ( x ) | − / Φ ˜ h ˜ D ( x ) on ˜ D , where ˜ h ( x ) = | ψ (cid:48) ( x ) | / h ( ψ ( x )) . Combining the scaling properties of Φ h with Theorem 5.4 leads to the precise power-law behavior of the mass m (Φ h ) of Φ h , defined as the supremum over all m such that,for all f, g ∈ C ∞ ( R ) and some C m ( f, g ) < ∞ ,(5.30) | Cov(Φ h ( f ) , Φ h ( T u g )) | ≤ C m ( f, g ) e − mu , where ( T u g )( x , x ) = g ( x − u, x ). Corollary 5.7 (Corollary 1.6 of [13]) . m (Φ h ) = Ch / for some C ∈ (0 , ∞ ) and all h . Remark 5.8.
As mentioned above, the power law behavior m (Φ h ) = Ch / of the mass m (Φ h ) follows from the scaling properties of the field Φ h . The fact that C ∈ (0 , ∞ )is equivalent to 0 < m (Φ h ) < ∞ , where the lower bound m (Φ h ) > m (Φ h ) = ∞ leads to a contradiction, which proves that m (Φ h ) < ∞ .We conclude this section and the paper with a sketch of the main ideas of the proofof the upper bound of Theorem 5.3. In the discussion below we assume that the readerhas some familiarity with (FK) percolation and we use extensively the FK representationof the Ising model described in Section 3, and in particular the couplings discussed inSection 3.3. In this context, the notation we use should be self-explanatory (e.g., we willuse { x ←→ y (cid:54)←→ g } to denote the event that vertices x and y are in the same FK clusterand that that cluster is not connected to the ghost). Figure 3.
An illustration of the event described in the sketch of the proofof Theorem 5.3 representing a chain of large FK clusters crossing a rectan-gle.The first step of the proof of exponential decay consists in writing (cid:104) σ x ; σ y (cid:105) β c ,H = P F Kβ c ,H ( x ←→ y ) − P F Kβ c ,H ( x ←→ g ) P F Kβ c ,H ( y ←→ g )(5.31) = P F Kβ c ,H ( x ←→ y (cid:54)←→ g ) + Cov F Kβ c ,H ( I { x ←→ g } , I { y ←→ g } ) , (5.32)where Cov F Kβ c ,H denotes covariance with respect to the FK measure P F Kβ c ,H on a Z .Letting B ( x, L ) denote the square centered at x of side length 2 L and writing A nearx := { there exists an FK-open path from x, within B ( x, | x − y | / , to some w with the edge from w to g open } and A farx := { x ←→ g } \ A nearx , so that { x ←→ g } = A nearx ∪ A farx , the covariance in(5.32) can be written as a sum of four covariances and (cid:104) σ x ; σ y (cid:105) β c ,H as a sum of five terms.Bounding four of these five terms reduces to showing that, when H = ha / ,(5.33) P F Kβ c ,H ( g (cid:54)←→ x ←→ ∂B ( x, | x − y | / ≤ ˜ C ( h ) a / e − ˆ C ( h ) | x − y | . The remaining term, Cov
F Kβ c ,H (1 A nearx , A neary ), needs a separate argument and will be dis-cussed later.Focusing for now on (5.33), the power law part of the upper bound comes from a 1-armargument, while the exponential part requires a more sophisticated argument that makesuse of CME / coupled to CLE / (see Section 4.1) as well as a stochastic dominationtheorem by Liggett, Schonmann and Stacey [38].Roughly speaking, what we use of the coupled CLE / and CME / is the fact that,for K large, a realization inside the rectangle Λ N,N := [0 , N ] × [0 , N ] is likely to containa chain of not more than K touching loops that cross the rectangle in the long direction,with the first loop touching one of the short sides of the rectangle and the last looptouching the opposite side (see Fig. 3). Moreover, the “area” of each of the continuumclusters associated to the loops in the chain is likely to be larger than N / /K , with theprobability of the event just described going to one as K → ∞ . Back on the lattice thisimplies that, inside Λ N,N , one can find with high probability a chain of FK clusters onelattice spacing away from each other and crossing the rectangle. Moreover, such clusterswill, with high probability, have sizes larger than N / a − / /K , which in turn impliesthat there is a high probability that they are each connected to g if the external field is H = ha / and N is large. ombining all of the above, with the help of the FKG inequality, one can show that,with high probability, a large annulus contains a circuit of FK clusters one lattice spacingaway from each other, each connected to g , such that the circuit disconnects the innersquare of the annulus from infinity. We call such an annulus good .In order to complete the proof of (5.33), one covers the plane with large overlappingannuli in such a way that their inner squares tile the plane. For each such annulus, theevent that it is good happens with high probability. We would like to conclude that goodannuli percolate, but the annuli are overlapping, so the events are not independent. Todeal with this, one can use a stochastic domination result due to Liggett, Schonmannand Stacey [38]. Now, percolation of good annuli implies that the probability that x issurrounded by a circuit of good annuli contained in a square B ( x, L ) of size 2 L centeredat x is close to one, exponentially in L . But because of planarity, if x is surrounded bya circuit of good annuli contained in B ( x, L ), the event { g (cid:54)←→ x ←→ ∂B ( x, L ) } cannothappen. This provides the desired exponential bound.The remaining term can be written asCov F Kβ c ,H ( I A nearx , I A neary ) = P F Kβ c ,H ( A nearx ∩ A neary ) − P F Kβ c ,H ( A nearx ) P F Kβ c ,H ( A neary )= P F Kβ c ,H ( A neary ) (cid:2) P F Kβ c ,H ( A nearx | A neary ) − P F Kβ c ,H ( A neary ) (cid:3) . A 1-arm argument provides a polynomial upper bound of order a / for P F Kβ c ,H ( A neary ). Thefirst step in dealing with the remaining factor consists in showing that P F Kβ c ,H ( A nearx | A neary )is smaller than the probability of the event A nearx with wired boundary condition on B ( x, | x − y | / B ( x, | x − y | /
3) with respectto the FK measure in B ( x, | x − y | /
3) with wired boundary condition. The remainingstep consists in showing that the probability of A cx is not affected much by the boundarycondition in B ( x, | x − y | / Acknowledgments.
The authors thank Wouter Kager for Figures 1 and 2, and ananonymous referee for useful comments. The first author thanks Fran¸cois Dunlop, EllenSaada, and Alessandro Giuliani for organizing the IRS 2020 conference and for the invita-tion to give a presentation and to contribute to the conference proceedings. The researchof the second author was partially supported by NSFC grant 11901394 and that of thethird author by US-NSF grant DMS-1507019.
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A.B. Zamolodchikov , Integrals of motion and S-matrix of the (scaled) T = T c Ising model withmagnetic field.
Int. J. Mod. Phys. Division of Science, NYU Abu Dhabi, Saadiyat Island, Abu Dhabi, UAE & Departmentof Mathematics, Faculty of Science, Vrije Universtiteit Amsterdam, De Boelelaan 1111,1081 HV Amsterdam, The Netherlands.
Email address : [email protected] Beijing Institute of Mathematical Sciences and Applications, No.11 Yanqi Lake WestRoad, Beijing 101407 & NYU-ECNU Institute of Mathematical Sciences at NYU Shang-hai, 3663 Zhongshan Road North, Shanghai 200062, China.
Email address : [email protected] Courant Institute of Mathematical Sciences, New York University, 251 Mercer st,New York, NY 10012, USA, & NYU-ECNU Institute of Mathematical Sciences at NYUShanghai, 3663 Zhongshan Road North, Shanghai 200062, China.
Email address : [email protected]@cims.nyu.edu