Quantitative disorder effects in low-dimensional spin systems
QQUANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPINSYSTEMS
PAUL DARIO, MATAN HAREL, RON PELED
Abstract.
The Imry–Ma phenomenon, predicted in 1975 by Imry and Ma and rigorouslyestablished in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are ‘rounded’ by the addition of a quenched random field to thequantity undergoing the transition. The phenomenon applies to a wide class of spin systems indimensions d ≤ d ≤ L , we study the effect of the boundary conditions on the spatial and thermalaverage of the quantity coupled to the random field. We show that the boundary effectdiminishes at least as fast as an inverse power of log log L for general two-dimensional spinsystems and for four-dimensional spin systems with continuous symmetry, and at least as fastas an inverse power of L for two- and three-dimensional spin systems with continuous symmetry.Specific models of interest for the obtained results include the two-dimensional random-field q -state Potts and Edwards-Anderson spin glass models, and the d -dimensional random-fieldspin O ( n ) models ( n ≥
2) in dimensions d ≤ Introduction
The large-scale properties of equilibrium statistical physics systems with quenched (frozen-in)disorder can differ significantly from those of the corresponding non-disordered systems [11,50, 51, 52]. The present understanding of such phenomena is still lacking, in both the physicaland mathematical literature, but some cases are better understood. This work focuses on theso-called Imry–Ma phenomenon by which first-order phase transitions of low-dimensional spinsystems are ‘rounded’ by the addition of a quenched random field to the quantity undergoingthe transition. Imry and Ma [40] studied the Ising and spin O ( n ) models and predicted that, inlow dimensions, the addition of a random independent magnetic field causes the systems to losetheir characteristic low-temperature ordered states, even when the strength of the added fieldis arbitrarily weak. Specifically, they predicted this effect for the random-field Ising model atall temperatures (including zero temperature) in dimensions d ≤
2, and for the random-fieldspin O ( n ) -model, with n ≥
2, at all temperatures in dimensions d ≤
4. Their predictionswere confirmed, and greatly extended, in the seminal work of Aizenman and Wehr [5, 6] whorigorously established the rounding phenomenon for a general class of spin systems in dimensions d ≤ d ≤ Z d , with aformal translation-invariant Hamiltonian H . Let η = ( η v ) v ∈ Z d be independent and identicallydistributed random variables (or random vectors). Construct a disordered spin system bymodifying the Hamiltonian H to H η,λ ( σ ) ∶= H ( σ ) − λ ∑ v ∈ Z d η v ⋅ f (T − v ( σ )) with T the translation operator defined by T v ( σ ) u = σ u − v , f the observable to which the randomfield η is applied, and λ the disorder strength. Aizenman and Wehr prove that, under suitableassumptions on H , η and f , in dimensions d ≤ a r X i v : . [ m a t h - ph ] J a n PAUL DARIO, MATAN HAREL, RON PELED strengths, the limit(1.1) lim L →∞ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f (T v ( σ ))⟩ µ exists with probability one (in terms of η ), and takes the same value for all infinite-volumeGibbs measures µ of the disordered system. The notation ⟨⋅⟩ µ denotes the thermal average inthe Gibbs measure µ ; it should be noted that, while the set of Gibbs measures of the disorderedsystem depends on η , Aizenman and Wehr further prove that the common limit (1.1) does notdepend on η . An analogous statement is proved to hold in dimensions d ≤ quantitative estimates on the rate of conver-gence in (1.1) and it is the goal of this work to obtain such a quantified result. The quantitythat we study is(1.2) sup τ ,τ RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f (T v ( σ ))⟩ τ Λ L − ⟨ f (T v ( σ ))⟩ τ Λ L RRRRRRRRRRR where ⟨⋅⟩ τ Λ is the thermal average under the finite-volume Gibbs measure (of the disorderedsystem) in the volume Λ ⊆ Z d with boundary conditions τ , and where Λ L ∶= {− L, . . . , L } d ⊆ Z d and ∣ Λ L ∣ is its cardinality. We are thus measuring the largest discrepancy in the value of thespatially- and thermally-averaged observable, which may arise when changing the boundaryconditions in finite volume. This quantity may be thought of as a particular kind of correlationdecay rate. We emphasize that the boundary conditions τ , τ appearing in (1.2) are allowed todepend on L and on the specific realization of the disorder η .Our main results are that the following holds with high probability over η : (1) For a generalclass of two-dimensional spin systems the quantity (1.2) decays at least as fast as an inversepower of log log L , (2) For a general class of spin systems with continuous symmetry, thequantity (1.2) decays at least as fast as an inverse power of L in dimensions d = , L in dimension d =
4. These results are shown to hold atall temperatures and positive disorder strengths.Quantitative estimates of the type obtained here have recently been developed for the two-dimensional random-field Ising model [21, 4, 29, 28, 3], where it was shown that correlationsdecay at an exponential rate (in the nearest-neighbor case) at all temperatures and positivedisorder strength; see Section 4 for more details. However, to our knowledge, this is the only spinsystem for which quantitative estimates have previously been developed and their derivationappears to crucially rely on monotonicity properties of the Ising model which are absent forgeneral spin systems. Specific examples of interest for the results obtained here include thetwo-dimensional random-field q -state Potts (with q ≥
3) and Edwards-Anderson spin glassmodels, and the d -dimensional random-field spin O ( n ) models (with n ≥
2) in dimensions d ≤ UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 3 glass model possesses a unique ground-state pair (see, e.g., [48, 47, 49, 7] and the referencestherein).The rest of the paper is structured as follows: Section 2 presents the setup and results forgeneral two-dimensional spin systems. The setup and results for spin systems with continuoussymmetry are presented in Section 3. Section 4 is devoted to an overview of related results. Anoutline of our proofs is given in Section 5. Section 6 collects preliminary properties of the spinsystems that we study. The proofs of the results pertaining to general two-dimensional spinsystems are presented in Section 7 while the proofs pertaining to spin systems with continuoussymmetry are presented in Section 8. Section 9 discusses additional points and highlights severalopen problems.
Acknowledgements.
We are grateful to Michael Aizenman, David Huse, Charles M. Newman,Thomas Spencer and Daniel L. Stein for encouragement and helpful conversations on the topicsof this work. We also thank Antonio Auffinger, Wei-Kuo Chen, Izabella Stuhl and Yuri Suhovfor the opportunity to present these results in online talks and for useful discussions. Theresearch of the authors was supported in part by Israel Science Foundation grants 861/15 and1971/19 and by the European Research Council starting grant 678520 (LocalOrder).2.
Two-dimensional disordered spin systems
In this section we describe the quantitative decay rates that are obtained for a general class oftwo-dimensional disordered spin systems. We first describe the spin systems to which the resultsapply (Section 2.1), then proceed to a list of specific examples which clarify and add interest tothe general definitions (Section 2.2), and finally describe the results themselves (Section 2.3).2.1.
The general setup.
We work on the standard integer lattice Z d , in which we denotethe standard orthonormal basis by e , . . . , e d ; we write u ∼ v if two vertices u, v are nearestneighbors. We let ∥ ⋅ ∥ ∞ denote the (cid:96) ∞ distance on Z d and, for v ∈ Z d and integer R ≥
0, set B ( v, R ) ∶= { w ∈ Z d ∶ ∥ w − v ∥ ∞ ≤ R } to be the ball of radius R around v in the (cid:96) ∞ distance.For integer R ≥
0, denote the external vertex boundary to distance R of a set Λ ⊆ Z d by ∂ R Λ ∶= { v ∈ Z d ∖ Λ ∶ B ( v, R ) ∩ Λ ≠ ∅} and set Λ + R ∶= Λ ∪ ∂ R Λ. We also abbreviate ∂ Λ ∶= ∂ Λand Λ + ∶= Λ + .Our general results (which do not rely on continuous symmetries) apply to disordered spinsystems defined via (2.3) and (2.4) below and built as follows. The base system:
A non-disordered spin system is constructed from the following elements.(1)
State space and configuration space:
Let (S , A , κ ) be a probability space. Configurations ofthe spin system are functions σ ∶ Z d → S . Their restriction to a subset Λ ⊆ Z d is denoted σ Λ .(2) Hamiltonian:
For each finite Λ ⊆ Z d , let H Λ ∶ S Z d → R be a bounded measurable map. Weassume that this family of Hamiltonians satisfies the following two properties:(a) Consistency:
For each finite Λ ⊆ Z d , inverse temperature β >
0, and boundary conditions τ ∶ Z d ∖ Λ → S , define the finite-volume Gibbs measure µ β, Λ ,τ ( dσ ) ∶= Z β, Λ ,τ exp (− βH Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) ∏ v ∈ Z d ∖ Λ δ τ v ( dσ v ) , where Z β, Λ ,τ , called the partition function, normalizes µ β, Λ ,τ to be a probability measure,and δ τ ( v ) is the Dirac delta measure at τ v . We denote by ⟨⋅⟩ τβ, Λ the expectation operatorwith respect to µ β, Λ ,τ . We omit β from the notation when it is clear from context.We assume that the family of Hamiltonians satisfies the following consistency relation(finite-volume Gibbs property): For each pair of finite subsets Λ ′ ⊆ Λ, each inversetemperature β >
0, each boundary condition τ ∶ Z d ∖ Λ → S , and each bounded PAUL DARIO, MATAN HAREL, RON PELED measurable g ∶ S Z d → R ,(2.1) ⟨⟨ g ( σ )⟩ τβ, Λ ′ ⟩ τ β, Λ = ⟨ g ( σ )⟩ τ β, Λ , where the spin σ in the left-hand side is distributed according to the measure µ β, Λ ′ ,τ ,and the boundary condition τ is random and is distributed according to the restrictionof µ β, Λ ,τ to the set Z d ∖ Λ ′ .(b) Bounded boundary effect:
We assume that there exists a constant C H ≥ ⊆ Z d ,(2.2) ∣ H Λ ( σ ) − H Λ ( σ ′ )∣ ≤ C H ∣ ∂ Λ ∣∣ Λ ∣ for σ, σ ′ ∶ Z d → S satisfying σ Λ = σ ′ Λ . The disordered system:
To form the disordered system from the base system, we add aterm of the form ∑ v ∈ Λ η v ⋅ f v ( σ ) to the Hamiltonian, where ( η v ) is a family of m -dimensionalrandom vectors (the quenched disorder) and ( f v ) is a family of m -dimensional functions of theconfiguration such that f v ( σ ) depends only on the restriction of σ to a neighborhood of v ofsome fixed radius R .(1) Disorder:
We let η = ( η v ) v ∈ Z d be a collection of independent standard m -dimensionalGaussian vectors; we use the symbols P and E to refer to the law and the expectationoperator with respect to η .(2) Noised observables:
Fix integers m ≥ R ≥
0. For each v ∈ Z d , we let f v ∶ S Z d → R m bea measurable function satisfyingBoundedness: ∣ f v ( σ )∣ ≤ σ ∈ S Z d , Finite range: f v ( σ ) = f v ( σ ′ ) when σ, σ ′ ∈ S Z d satisfy σ B ( v,R ) = σ ′ B ( v,R ) . (3) Disordered Hamiltonian:
Given a finite Λ ⊆ Z d , a fixed realization of η ∶ Z d → R and adisorder strength λ >
0, define the disordered Hamiltonian H η,λ Λ ∶ S Z d → R by(2.3) H η,λ Λ ( σ ) ∶= H Λ ( σ ) − λ ∑ v ∈ Λ η v ⋅ f v ( σ ) (where the dot product denotes the Euclidean scalar product on R m ) and, for an inversetemperature β > τ ∈ S Z d ∖ Λ , define the finite-volume Gibbs measure(2.4) µ η,λβ, Λ ,τ ( dσ ) ∶= Z η,λβ, Λ ,τ exp (− βH η,λ Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) ∏ v ∈ Z d ∖ Λ δ τ v ( dσ v ) , where Z η,λβ, Λ ,τ is the normalization constant which makes the measure µ η,λβ, Λ ,τ a probabilitymeasure. We denote by ⟨⋅⟩ τ,η,λβ, Λ the expectation with respect to the measure µ η,λβ, Λ ,τ and referto it as the thermal expectation . When β, η and λ are clear from the context, we will omitthem from the notation.We note that the consistency relation (2.1) implies the same identity for the disorderedsystem: Given a pair of finite subsets Λ ′ ⊆ Λ, η ∶ Z d → R , λ > β > τ ∈ S Z d ∖ Λ and anybounded measurable g ∶ S Z d → R ,(2.5) ⟨⟨ g ( σ )⟩ τ Λ ′ ⟩ τ Λ = ⟨ g ( σ )⟩ τ Λ , where σ in the left-hand side is distributed as µ η,λβ, Λ ′ ,τ , and τ is random and distributedaccording to the restriction to Z d ∖ Λ ′ of a random spin configuration distributed as µ η,λβ, Λ ,τ .We point out that neither the base system nor the noised observables ( f v ) are requiredto be periodic with respect to translations. Still, it is very natural to work in a translation-invariant setup , by which we mean that for all v ∈ Z d and configurations σ we have (i) H Λ ( σ ) = H Λ + v (T v ( σ )) for all finite Λ, where T v is the translation by v operation: T v ( σ ) u = σ u − v , UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 5 and (ii) f v ( σ ) = f (T − v ( σ )) (where is the zero vector in Z d ). Indeed all of the examplespresented in the next section are of this type; however, we stress that such invariance is notrequired for our results (see Section 2.3). The additional flexibility of the general definitionsallows for inhomogeneities in the base system and further allows to vary the noised observables ( f v ) periodically along a sublattice of Z d (this may make sense, e.g., in antiferromagneticsystems, where the ordered state of the base system is not invariant to all translations), or evento choose ( f v ) arbitrarily.2.2. Examples.
We now describe several classical examples of disordered systems which fitwithin the general class of systems discussed in the previous section. To the best of ourknowledge, our general results are already new when specialized to these examples except forthe example of the random-field ferromagnetic Ising model where stronger results are known(see Section 4). ● Random-field Ising model:
The state space is
S ∶= {− , } equipped with the counting measureand the Hamiltonian of the base system (the ferromagnetic Ising model) is H Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v σ u σ v − ∑ v ∈ Λ hσ v , for a fixed h ∈ R . The noised observable is f v ( σ ) ∶= σ v so that m = R = H η,λ Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v σ u σ v − ∑ v ∈ Λ ( λη v + h ) σ v corresponding to the addition of a quenched random field to the Ising model. One similarlyforms the random-field antiferromagnetic Ising model by removing the minus sign in front ofthe term ∑ σ u σ v . ● Random-field q -state Potts model: Let q ≥ S ∶= { , . . . , q } equipped with the counting measure and the Hamiltonian of the base system (the ferromagnetic q -state Potts model) is H Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v { σ u = σ v } − ∑ v ∈ Λ q ∑ k = h k { σ v = k } , for a fixed h ∶= ( h , . . . , h q ) ∈ R q . The noised observable is f v ( σ ) ∶= ( { σ v = } , . . . , { σ v = q } ) sothat m = q , R = H η,h,λ Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v { σ u = σ v } − ∑ v ∈ Λ q ∑ k = ( λη v,k + h k ) { σ v = k } where we write η v = ( η v, , . . . , η v,q ) ∈ R q and h = ( h , . . . , h q ) ∈ R q . The effect of the disorder isto add an energetic bonus or penalty to each spin state at each vertex according to quenchedrandom vectors of length q which are assigned independently to the vertices. ● Edwards-Anderson spin glass : The state space is
S ∶= {− , } equipped with the countingmeasure and the Hamiltonian of the base system (a uniform system) is H Λ ( σ ) ∶= . The noised observable is f v ( σ ) ∶= ( σ v σ v + e i ) i ∈{ ,...,d } so that m = d , R = η assigned to every edge of Z d . The disordered Hamiltonian is (up to a termwhich does not depend on the spins in Λ) H η,h,λ Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v ( λη u,v + h ) σ u σ v PAUL DARIO, MATAN HAREL, RON PELED corresponding to adding a random energetic bonus/penalty to satisfied edges (edges with thesame spin state assigned to both endpoints) independently between the different edges. ● Random-field spin O ( n ) -model: Let n ≥ S ∶= S n − ⊆ R n equipped with the uniform measure and the Hamiltonian of the base system (the spin O ( n ) model) is H Λ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v σ u ⋅ σ v . The noised observable is f v ( σ ) = σ v ∈ R n so that R = m = n and the disordered Hamiltonianis H η,h, ΛΛ ( σ ) ∶= − ∑ { u,v }∩ Λ ≠∅ u ∼ v σ u ⋅ σ v − ∑ v ∈ Λ ( λη v + h ) ⋅ σ v corresponding to the addition of a quenched random field in a random direction, independentlyat each vertex, to the spin O ( n ) -model.We point out that while the results of the next section apply to the random-field spin O ( n ) -model, stronger results are obtained for it in Section 3 by relying on its continuoussymmetry (at h = Results.
Throughout the section, we fix a disordered spin system of the type defined inSection 2.1, described by a state space (S , A , κ ) , a family of Hamiltonians for the base system ( H Λ ) satisfying the bounded boundary effect assumption with constant C H and a family ofobservables ( f v ) with a range of R , taking values in R m . The disorder ( η v ) is a collection ofindependent standard m -dimensional Gaussian vectors. The results of this section apply totwo-dimensional spin systems so we further fix d = L ∶= {− L, . . . , L } d ⊆ Z d , let ∣ Λ L ∣ be its cardinality, and denote by ∣⋅∣ the Euclidean norm on R m . Theorem 1.
Let β > be the inverse temperature and λ > be the disorder strength. Thereexist constants C, c > depending only on λ , C H , m and R such that, for each integer L ≥ , (2.6) P ⎛⎜⎝ sup τ ,τ ∈S Z d ∖ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L RRRRRRRRRRR > C √ ln ln L ⎞⎟⎠ ≤ exp (− cL ) and moreover, in a translation-invariant setup, (2.7) P ⎛⎝ sup τ ∈S Z d ∖ Λ L RRRRRRRRRRR α − ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f (T − v σ )⟩ τ Λ L RRRRRRRRRRR > C √ ln ln L ⎞⎠ ≤ exp (− cL ) where α ∈ R m depends only on the spin system considered, on the inverse temperature β and onthe disorder strength λ (but does not depend on the disorder η and on L ). We mention that, for any fixed L , any v ∈ Λ L , and any boundary condition τ , the function η ↦ ⟨ f v ( σ )⟩ τ Λ L is Lipschitz continuous, uniformly in v and τ . Thus, the quantities inside theprobabilities in the previous theorems and the following ones are indeed measurable.We remark that, while we only prove the result at positive temperature, the same argumentyields the following zero-temperature version of the theorem: for a given side length L , definean L -ground configuration as a configuration σ ∶ Z d → S satisfying H η,λ Λ L ( σ ) = inf σ ′ ∶ Z d →S σ ′ ≡ σ in Z d ∖ Λ L H η,λ Λ L ( σ ′ ) , UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 7 and let G L be the set of L -ground configurations. Then we have(2.8) P ⎛⎝ sup σ ,σ ∈ G L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L f v ( σ ) − f v ( σ )RRRRRRRRRRR > C √ ln ln L ⎞⎠ ≤ exp (− cL ) where if G L is empty then the condition in the probability is considered to hold. Additionally, ifwe assume that the state space is compact and both the base Hamiltonian and noised observablesare continuous, we can show that, for all η and all L , the set G L of L -ground configurations isnon-empty. It is straightforward that this is the case in all the example systems of Section 2.2.In the translation invariant setup, the value α is explicit for some of the models describedin Section 2.2: for the q -state Potts model with external field h =
0, one has α = ( / q, . . . , / q ) .In the case of the Edwards-Anderson spin glass, a gauge symmetry (obtained by flipping thespins on the even sublattice and changing the sign of η on all edges) implies that the density ofsatisfied edges is about 1 /
2, i.e.,lim L →∞ ∣ E ( Λ L )∣ ∑ x,y ∈ Λ L x ∼ y ⟨ { σ x = σ y } ⟩ τ L Λ L —→ L →∞ P -almost-surely . for any collection of random boundary condition η ↦ τ L ( η ) ∈ S Z d ∖ Λ L and L ≥
1, where ∣ E ( Λ L )∣ denotes the number of edges in the box Λ L . In both these examples, our results are novel.2.3.1. Uniqueness conjecture and additional results.
Theorem 1 states that the spatial averageover Λ L of the thermal expectations ⟨ f v ( σ )⟩ τ Λ L does not depend strongly on the boundarycondition τ . It is natural to ask for a more detailed result: to what extent can the value of ⟨ f v ( σ )⟩ τ Λ L at a given vertex v be affected by τ ? Can the values at specific vertices v not tooclose to the boundary of Λ L , or even at most v ∈ Λ L , be significantly altered by changing τ (keeping in mind that the overall spatial average does not depend strongly on τ in the sense ofTheorem 1)? We conjecture that such a phenomenon cannot occur. We first state this as aconjecture, and then explain additional motivation as well as additional rigorous support (weremind that throughout the section we have fixed a disordered spin system of the type definedin Section 2.1 and we work in dimension d = Conjecture 2.1.
Let β > λ > P -almost-surely,(2.9) lim L →∞ sup τ ,τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ∣⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L ∣ = P -almost-surely,(2.10) lim L →∞ sup τ ,τ ∈S Z d ∖ Λ L ∣⟨ f ( σ )⟩ τ Λ L − ⟨ f ( σ )⟩ τ Λ L ∣ = . We make several remarks about the conjecture:In the translation-invariant setup, the pointwise statement (2.10) implies the averagedstatement (2.9). This can be argued, for instance, by applying Birkhoff’s ergodic theorem tothe functions g (cid:96) ( η ) ∶= sup τ ,τ ∈S Z d ∖ Λ (cid:96) ∣⟨ f ( σ )⟩ τ Λ (cid:96) − ⟨ f ( σ )⟩ τ Λ (cid:96) ∣ . We also note that the pointwise statement (2.10) may be reformulated as a statement on the( η -dependent) set of Gibbs measures of the disordered system. Indeed, let us assume that, inaddition to having a translation-invariant setup, the state space S is Polish and compact, with A the Borel sigma algebra, and the noised observables f v are continuous. These assumptionsallow to extract a subsequential limiting Gibbs state µ out of a sequence of finite-volume Gibbsstates µ k on increasing domains so that ⟨ f ( σ )⟩ µ k → ⟨ f ( σ )⟩ µ (where ⟨⋅⟩ µ is the expectation PAUL DARIO, MATAN HAREL, RON PELED under µ ). Then, the statement (2.10) is equivalent to the claim that, at any β, λ >
0, it holds P -almost surely that ⟨ f ( σ )⟩ µ takes the same value for all Gibbs measures µ of the disorderedsystem.In monotonic systems such that m = τ min , τ max suchthat ⟨ f v ( σ )⟩ τ min Λ L ≤ ⟨ f v ( σ )⟩ τ Λ L ≤ ⟨ f v ( σ )⟩ τ max Λ L for all τ and v (such as the plus and minus boundaryconditions in the random-field ferromagnetic Ising model), the averaged statement (2.9) followsimmediately from Theorem 1.Another statement which would be of interest to prove, and which is implied by the averagedstatement (2.9), is the following: P -almost-surely,(2.11) lim (cid:96) →∞ lim L →∞ sup τ ,τ ∈S Z d ∖ Λ L ∣ Λ (cid:96) ∣ ∑ v ∈ Λ (cid:96) ∣⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L ∣ = τ , τ ∈ S Z d ∖ Λ L by the supremum over σ , σ ∈ G L , where G L is the set of L -ground configurations and replace ⟨ f v ( σ )⟩ τ i Λ L by f v ( σ i ) .The zero-temperature pointwise statement would imply that, P -almost-surely, all groundconfigurations σ (i.e., all σ ∈ ∩ L G L ) agree on f v ( σ ) for all v ∈ Z d . In the specific case ofthe Edwards-Anderson spin glass model this is the same as the well-known belief that thetwo-dimensional model has a unique ground-state pair.One way in which typical configurations of the system can fulfill Theorem 1 but avoid theuniqueness statement in (2.9) is if the configurations are differently ordered on the two bipartiteclasses of Z (by the two bipartite classes, we mean the vertices with even sum of coordinates andthe vertices with odd sum of coordinates). Such a situation is familiar from the (non-disordered)antiferromagnetic Ising model at low temperature, in which typical configurations have achessboard-like pattern (and the boundary conditions can decide which of the two chessboardpatterns will emerge). Our next result shows, as a special case, that such behavior cannot arisefor the disordered systems considered herein, by showing that the boundary conditions cannotsignificantly influence the average value of ⟨ f v ( σ )⟩ on any deterministic set of positive density. Theorem 2.
Let β > be the inverse temperature and λ > be the disorder strength. Thereexist constants C, c > depending only on λ , C H , m and R such that for each integer L ≥ andfor each weight function w ∶ Λ L → [− , ] m , (2.12) P ⎛⎜⎝ sup τ ,τ ∈S Z d ∖ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L w ( v ) ⋅ (⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L )RRRRRRRRRRR ≤ C √ ln ln L ⎞⎟⎠ ≥ − exp (− cL ) . A second motivation for Theorem 2 comes from Parseval’s identity, which, in our setting,reads(2.13) 1 ∣ Λ L ∣ ∑ v ∈ Λ L ∣⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L ∣ = ∑ k ∈ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L e πi k ⋅ v L + (⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L )RRRRRRRRRRR . Consequently, if one can upgrade the statement of Theorem 2 and obtain a rate of convergencefaster than 1 / L (instead of the ( log log L ) − / term), then it would imply that the right-handside of (2.13) is small, which would then yield (2.9).As a corollary of Theorem 2, we obtain a quantitative estimate in the spirit of the averagedstatement (2.9). The obtained result is weaker than (2.9) as it involves the expected value (inthe random field) of ⟨ f v ( σ )⟩ τ ( η ) Λ L . Corollary 2.2.
Let β > be the inverse temperature and λ > be the disorder strength. Thereexist constants C, c > depending only on λ , C H , m and R such that, for each integer L ≥ UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 9 and each random (i.e., measurable) pair of boundary conditions η ↦ τ ( η ) , τ ( η ) ∈ S Z d ∖ Λ L , (2.14) 1 ∣ Λ L ∣ ∑ v ∈ Λ L ∣ E [⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L ]∣ ≤ C √ ln ln L .
As before, versions of Theorem 2 and Corollary 2.2 hold at zero temperature. In Theorem 2the supremum over boundary conditions should be replaced by a supremum over σ , σ ∈ G L and ⟨ f v ( σ )⟩ τ i Λ L should be replaced by f v ( σ ) . In Corollary 2.2 the random boundary conditionsshould be replaced by random σ , σ ∈ G L and ⟨ f v ( σ )⟩ τ i ( η ) Λ L should be replaced by f v ( σ ) .3. Spin systems with continuous symmetry
In this section we present quantitative results for spin systems with continuous symmetry.The class of systems that we study are versions of the random-field spin O ( n ) model and aredescribed similarly to Section 2.1 with the following additional assumptions:(1) State space:
We let S be the sphere S n − for some integer n ≥
1, equipped with its Borelsigma algebra and the uniform measure κ .(2) Base Hamiltonian:
We assume that, for any finite subset Λ ⊆ Z d , the Hamiltonian H Λ takesthe form(3.1) H Λ ( σ ) = ∑ { u,v }∩ Λ ≠∅ u ∼ v Ψ ( σ u , σ v ) , where the map Ψ ∶ S n − × S n − → R is twice continuously differentiable and rotationallyinvariant: for any R ∈ O ( n ) and any σ , σ ∈ S n − , Ψ ( Rσ , Rσ ) = Ψ ( σ , σ ) .(3) The noised observable:
We assume that m = n , and that f v ( σ ) = σ v . In particular theobservables have range R = Disordered Hamiltonian:
It will turn out to be useful to incorporate a deterministic externalmagnetic field h ∈ R n in the disordered Hamiltonian. Thus, given additionally a finiteΛ ⊆ Z d , a fixed realization of η ∶ Z d → R and a disorder strength λ >
0, we define(3.2) H η,λ,h Λ ( σ ) ∶= H Λ ( σ ) − ∑ v ∈ Λ ( λη v + h ) ⋅ σ v . As we work with nearest-neighbor systems, it suffices to specify the boundary conditionon the external boundary ∂ Λ of the domain Λ and we will do so in the sequel (instead ofspecifying the boundary condition on Z d ∖ Λ). Given an inverse temperature β > τ ∈ S ∂ Λ , we define the finite-volume Gibbs measure(3.3) µ η,λ,hβ, Λ ,τ ( dσ ) ∶= Z η,λβ, Λ ,τ exp (− βH η,λ,h Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) ∏ v ∈ Z d ∖ Λ δ τ v ( dσ v ) , where Z η,λ,hβ, Λ ,τ is the normalization constant which makes the measure µ η,λ,hβ, Λ ,τ a probabilitymeasure. We denote by ⟨⋅⟩ τ,h Λ the expectation with respect to the measure µ η,λ,hβ, Λ ,τ (omittingthe parameters β, η and λ in the notation as was done in Section 2).We remark that the setting is sufficiently flexible to include periodic boundary conditions(i.e. taking Λ to be the discrete torus), as well as free boundary conditions. These arereferred to by the notations per and free.The following results show that, in dimensions 1 ≤ d ≤
3, the spatially- and thermally-averaged magnetization is close to zero when the external field h is close to zero, uniformly inthe boundary condition. The results depend on L through power laws. The first statementapplies to deterministic boundary conditions, and has a better exponent in the power law whencompared with the second statement, which is uniform in the (possibly random) boundaryconditions (see Theorem 5 for more details on the case that h is large).We use the notation a ∨ b ∶= max ( a, b ) and a ∧ b ∶= min ( a, b ) for a, b ∈ R . Theorem 3.
Let n ≥ , d ∈ { , , } and L ≥ . Let β > be the inverse temperature, λ > bethe disorder strength and h ∈ R n be the deterministic external field. There exists a constant C > depending only on n , Ψ and λ such that when h satisfies ∣ h ∣ ≤ L − , then for each boundarycondition τ ∈ S ∂ Λ L (allowing also free and periodic boundary conditions), (3.4) RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ CL − ( − d ) and (3.5) E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ,h Λ L RRRRRRRRRRR⎤⎥⎥⎥⎥⎦ ≤ CL − − d − d . The next result presents our quantitative estimates in dimension d = Theorem 4.
Let n ≥ and d = . Let β > be the inverse temperature, λ > be the disorderstrength and h ∈ R n with ∣ h ∣ ≤ be the deterministic external field. There exist constants C > depending only on n , Ψ and λ such that for each integer L ≥ , (3.6) E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ,h Λ L RRRRRRRRRRR⎤⎥⎥⎥⎥⎦ ≤ C √ ln ln (∣ h ∣ − ∧ L ) . We again remark that, in the second part of Theorem 3 and Theorem 4, free and periodicboundary conditions are allowed as one of the options in the supremum.We mention that versions of Theorem 3 and 4 hold also at zero temperature. To state theresults, we introduce the following notations. Given L ∈ N and a boundary condition τ ∈ S ∂ Λ L ,we define the zero-temperature free energy byFE τ,λ,h Λ ,T = ( η ) ∶= inf σ ′ ∶ Λ + L →S σ ′ ≡ τ on ∂ Λ L H η,λ,h Λ ( σ ′ ) . We then define the set G τ,ηL of L -ground configurations with boundary condition τ as the set ofconfigurations σ ∶ Λ + L → S satisfying σ ≡ τ on ∂ Λ L and H η,λ,h Λ ( σ ) = FE τ,λ,h Λ ,T = ( η ) . The continuity of the disordered Hamiltonian and the compactness of the spin space ensure thatthe set G τ,ηL is non-empty for all realizations of the random field η . Additionally, we remark thatthe cardinality of the set G τ,ηL is almost surely equal to 1. Indeed, this result is a consequence ofthe following observation: since the map η ↦ FE τ,λ,h Λ ,T = ( η ) is concave, it is differentiable almosteverywhere, and, on the corresponding set of full measure, the L -ground configuration withboundary condition τ is uniquely characterised as − λ times the η -gradient of FE τ,λ,h Λ ,T = .The zero-temperature version of the results is then obtained by replacing the mapping η ↦ ⟨ σ v ⟩ τ,h Λ L in (3.4) by the map η ↦ σ v with σ ∈ G τ,η L . In (3.5) and (3.6), the supremumover boundary conditions should be replaced by a supremum over all L -ground configurations σ ∈ ∪ τ ∈S ∂ Λ L G τ,ηL and ⟨ σ v ⟩ τ,h Λ L should be replaced by σ v .Lastly, we mention that a discussion of models with continuous symmetries of higher orderappears in Section 9. 4. Background
This section presents a brief overview of related results.
The random-field Ising model:
The question of the quantification of the Imry-Ma phenomenonwas previously addressed for the two-dimensional random-field Ising model. Chatterjee [21]obtained an upper bound at the rate ( ln ln L ) − / on the effect of boundary conditions on the UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 11 magnetization at the center of a box of side length L . Aizenman and the third author [4] studiedthe same quantity for finite-range random-field Ising models and obtained an algebraic upperbound of the form L − γ . Exponential decay was established first at high temperature or strongdisorder; results in this direction include the ones of Fr¨ohlich and Imbrie [33], Berreti [10], VonDreifus, Klein and Perez [56], and Camia, Jiang and Newman [20]. Exponential decay at alldisorder strengths was established for the nearest-neighbor model by Ding and Xia [29] at zerotemperature, and then extended to all temperatures by Ding and Xia [28] and by Aizenman,the second and third authors [3].Bricmont and Kupiainen [16] introduced a hierarchical approximation to the random-fieldIsing model and proved that it exhibits spontaneous magnetization in three dimensions and nomagnetization in two dimensions. Imbrie [39] proved that the three-dimensional ground state ofthe Ising model in a weak magnetic field exhibits long range order, and this was extended to thelow-temperature regime by Bricmont and Kupiainen [15, 17] using a rigorous renormalizationgroup argument.Recently in [14], Bowditch and Sun constructed a continuum version of the two-dimensionalrandom-field Ising model by considering the scaling limit of the suitably renormalized discreterandom-field Ising model, and proved that the law of the limiting magnetization field is singularwith respect to the one of the continuum pure two-dimensional Ising model constructed andstudied by Camia, Garban and Newman [18, 19].Additional background on the random field Ising model can be found in [11, Chapter 7]and [46]. Random-field induced order in the XY -model: Crawford [24, 25, 26] studied the XY -modelin the presence of a random field pointing along the Y -axis, and proved that this may lead tothe model exhibiting residual ordering along the X -axis. We additionally refer to the referencestherein for a review of the works in the physics literature investigating this phenomenon indifferent models of statistical physics. Other models of statistical physics:
The Imry–Ma phenomenon has also been studied in thecontext of random surfaces. This line of investigation was initiated by Bovier and K¨ulske [12,13] with later, qualitative and quantitative, contributions of Cotar, van Enter, K¨ulske andOrlandi [43, 44, 55, 22, 23]. Additional quantitative results are obtained in [27] which studies“random-field random surfaces” of the form(4.1) P ( dφ ) ∶= Z exp ⎛⎝− ∑ x ∼ y V ( φ x − φ y ) + λ ∑ x η x φ x ⎞⎠ ∏ x dφ x , where in one case φ is a mapping defined on the lattice and valued in R , the map V ∶ R → R isa uniformly convex potential, and dφ x denotes the Lebesgue measure and in another case φ is valued in Z , V ( x ) = x and dφ x denotes the counting measure. We refer to [27] for a moredetailed review of this line of investigation.A version of the rounding effect was also studied in models of directed polymers by Giacominand Toninelli [36, 35]. They established that, while the non-disordered model may exhibit a first-(or higher-) order phase transition, this transition is at least of second order in the presence of arandom disorder. Nevertheless, the mechanism taking place is of different nature than the oneunderlying the Imry–Ma phenomenon [36]. Quantum systems:
The arguments developed in [6] were extended by Aizenman, Greenblattand Lebowitz [2] to quantum lattice systems. They established that, at all temperatures, thefirst-order phase transition of these systems is rounded by the addition of a random field indimensions d ≤
2, and in every dimensions d ≤ Strategy of the proof
In this section, we outline the proofs of Theorem 1, Theorem 3 and Theorem 4, which containthe main ideas developed in the article. To simplify the presentation of the arguments, weassume that m = λ of the random field is equal to 1, and that the maps ( f v ) have range 0 (i.e., f v depends only on the the value of the spin σ v ).5.0.1. Outline of the proof of Theorem 1.
The proof relies on a thermodynamic approach andrequires the introduction of the finite-volume free energy of the system as follows: for each sidelength L ≥
2, each boundary condition τ ∈ S Z d ∖ Λ L , inverse temperature β >
0, and magnetic field η ∶ Λ L → R , we define the finite-volume free energy to be the suitably renormalized logarithm ofthe partition function(5.1) FE τ Λ L ( η ) ∶= − β ∣ Λ L ∣ ln Z η,λβ, Λ L ,τ . The proofs then rely on the following standard observations: ● By decomposing the random field according to η = ( ˆ η L , η ⊥ L ) with ˆ η L ∶= ∣ Λ L ∣ − ∑ v ∈ Λ L η v and η ⊥ L ∶= η − ˆ η L , we see that the the observable ∣ Λ L ∣ − ∑ v ∈ Λ L ⟨ f v ( σ )⟩ τ Λ L can be characterizedas the derivative of the free energy with respect to the averaged external field ˆ η L , i.e., ∂∂ ˆ η L FE τ Λ L ( η ) = − ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f v ( σ )⟩ τ Λ L . ● For each realization of η ⊥ L , the mapping ˆ η L ↦ − FE τ Λ L ( ˆ η L , η ⊥ L ) is convex, differentiableand 1-Lipschitz. Moreover, for any pair of boundary conditions τ , τ ∈ S Z d ∖ Λ L , thefinite-volume free energy satisfies the relation(5.2) ∣ FE τ Λ L ( η ) − FE τ Λ L ( η )∣ ≤ CL . ● The averaged external field ˆ η L is a Gaussian random variable whose variance is equal to ∣ Λ L ∣ − . In two dimensions, its fluctuations are of order L − , which is the same order ofmagnitude as the right-hand side of (5.2).The key input of the argument is the following general fact: for each 1-Lipschitz convex anddifferentiable function g ∶ R → R and each δ >
0, there exists a set A g ⊆ R satisfying the twofollowing properties:(1) The Lebesgue measure of the set R ∖ A finite and satisfies Leb ( R ∖ A ) ≤ C / δ .(2) For each point x ∈ A , and each differentiable, 1-Lipschitz and convex function g ∶ R → R satisfying sup t ∈ R ∣ g ( t ) − g ( t )∣ ≤
1, one has ∣ g ′ ( x ) − g ′ ( x )∣ ≤ δ. This observation is quantified through the notion of δ -stability introduced in Section 7.1.Applying this property to the free energies ˆ η L ↦ FE τ Λ L ( ˆ η L , η ⊥ L ) , for a fixed boundary condi-tion τ ∈ S Z d ∖ Λ L , using the inequality (5.2), that the averaged field ˆ η Λ L is Gaussian and that itsvariance is of order L − , we obtain, for any δ > P ⎛⎜⎝ sup τ ,τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L < δ ⎞⎟⎠ ≥ c δ , where the constant c δ depends only on δ and satisfies c δ ≥ e − C / δ .The lower bound (5.3) is weaker than the statement of Theorem 1. The strategy to upgradethe inequality (5.3) into the quantitative estimate (2.6) relies on the observation that the UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 13 inequality (5.3) is scale invariant in two dimensions (the constant c δ does not depend on L ).One can thus implement a hierarchical decomposition of the space and leverage on this propertyto improve the result. Specifically, we implement a Mandelbrot percolation argument whichallows to cover (almost completely) the box Λ L by a collection Q of disjoint boxes (see Figure 2),all of which satisfying the property(5.4) sup τ ,τ ∈S Z d ∖ Λ ∣ Λ L ∣ ∑ v ∈ Λ ⟨ f v ( σ )⟩ τ Λ − ⟨ f v ( σ )⟩ τ Λ ≤ C √ ln ln L .
Once this is achieved, an application of the domain subadditivity property for the left-handside of (5.4) (see Section 6.4) yields the bound(5.5) P ⎛⎜⎝ sup τ ,τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f v ( σ )⟩ τ Λ L − ⟨ f v ( σ )⟩ τ Λ L ≥ C √ ln ln L ⎞⎟⎠ ≤ exp (− c √ ln L ) . The inequality (5.5) is slighlty weaker than Theorem 1, since the stochastic integrability (theright-hand side of (5.5)) can be improved. This is achieved by using a concentration argument.We complete this outline with a remark regarding the rate of convergence obtained: it isrelated to the lower bound c δ through the formulainf { δ ∈ ( , ) ∶ c δ ∶= L } . The result of Lemma 7.5 gives the value c δ ∶= exp (− Cδ ) , which yields the rate 1 / √ ln ln L .5.0.2. Outline of the proofs of Theorem 3 and Theorem 4.
To simplify the presentation of theargument, we make the additional assumption assume that h =
0. The proofs of Theorem 3 andTheorem 4 rely on a Mermin-Wagner argument (see [45]) which allows to use the continuoussymmetry of the model to upgrade the inequality (5.2): the upper bound we obtain is stated inProposition 8.4 and reads(5.6) E [ FE τ, L (̃ η ) − FE τ, L ( η ) ∣ η Λ Λ L / ] ≤ CL , where the free energy is defined in (6.9) below, the expectation in the left-hand side of (5.6) isthe conditional expectation with respect to the values of the random field η inside the box Λ L / (see Section 6.3), and the field ̃ η is defined by ̃ η v = − η v if v ∈ Λ L / and ̃ η v = η v if v ∉ Λ L / . Themain features of the inequality (5.6) are the following: ● The right-hand side of (5.6) decays like C / L , in contrast to the upper bound C / L of (5.2) in the case of general spin systems. ● The derivative with respect to the averaged field ˆ η Λ L / ∶= ∣ Λ L / ∣ − ∑ v ∈ Λ L / η v of theleft-hand side of (5.6) is explicit and satisfies the equality(5.7) ∂∂ ˆ η L / E [ FE τ, L ( η ) − FE τ, L (̃ η ) ∣ η Λ Λ L / ] = − ∣ Λ L ∣ ∑ v ∈ Λ L / E [⟨ σ v ⟩ τ, L ( η ) + ⟨ σ v ⟩ τ, L (̃ η ) ∣ η Λ Λ L / ] . In particular, using the η → − η invariance of the law of the random field, we see thatthe expectation of the right-hand side of (5.7) is equal to 2 E [∣ Λ L ∣ − ∑ v ∈ Λ L ⟨ σ v ⟩ τ, L ] .The expectation of the spatially and thermally-averaged magnetization can thus becharacterized as the expectation of the derivative with respect to the variable ˆ η Λ L / ofthe left-hand side of (5.6).In dimensions d ≤
3, the fluctuations of the averaged field ˆ η Λ L / are of order L − d / , and arethus larger than the right-hand side of (5.6). Combining this observation with a variationalprinciple (see Section 8.1) yields the algebraic rate of convergence stated in Theorem 3. In dimension 4, the fluctuations of the averaged field ˆ η Λ L / are of the same order of magnitude asthe right-hand side of (5.6), and we implement a Mandelbrot percolation argument similar tothe one presented in Section 5.0.1 to obtain the result.We point out that there is a distinct difference between the two constructions. In the presenceof a continuous symmetry, we do not rely on the criterion (5.4) to define which box shouldbelong to the partition Q but rather on the inequality, for a box Λ ⊆ Λ L , ∣ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v ⟩ τ, L ∣ ˆ η Λ ]∣ ≤ C √ ln ln L , where the conditional expectation is taken with respect to the averaged field ˆ η Λ ∶= ∣ Λ ∣ − ∑ v ∈ Λ η v .This difference accounts for the slightly better rate of convergence obtained in Theorem 4: weobtain the rate 1 /√ ln ln L instead of the rate 1 / √ ln ln L in Theorem 1.6. Notations, assumptions and preliminaries
General notation.
A box is then a subset of the form v + Λ L for v ∈ Z d and L ∈ N . Wecall the vertex v the center of the box Λ and the integer ( L + ) its side length. Given a boxΛ ⊆ Z d of side length L and a real number α >
0, we denote by α Λ the box with the same centeras Λ and side length ⌊ αL ⌋ , and define Λ αL ∶= α Λ L . Given a set Λ ⊆ Z d and a vertex v ∈ Z d ,we denote by dist ( v, Λ ) ∶= min w ∈ Λ ∣ v − w ∣ . For each set Λ ⊆ Z d , we denote by 1 Λ the indicatorfunction of Λ.For each real number a ∈ R , we use the notation ⌊ a ⌋ to refer to the integer part of a .Given a function g defined on either R , a subset Λ ⊆ Z d or the set of configurations andvalued in R m , we denote by g , . . . , g m its components. In particular, we denote by f v, , . . . , f v,m the components of the observable f v , and, in the case of continuous spin systems studied inSection 8, we denote by σ v, , . . . , σ v,n the components of the spin σ v .For each bounded set Λ ⊆ Z d and each fixed boundary condition τ ∈ S Z d ∖ Λ , we denote by(6.1) Fluc Λ ( η ) ∶= sup τ ,τ ∈S Z d ∖ Λ ∣ ∣ Λ ∣ ∑ v ∈ Λ ⟨ f v ( σ )⟩ τ Λ − ⟨ f v ( σ )⟩ τ Λ ∣ , and(6.2) Fluc τ Λ ( η ) ∶= sup τ ∈S Z d ∖ Λ ∣ ∣ Λ ∣ ∑ v ∈ Λ ⟨ f v ( σ )⟩ τ Λ − ⟨ f v ( σ )⟩ τ Λ ∣ . Similarly, for each i ∈ { , . . . , m } , we define(6.3) Fluc Λ ,i ( η ) ∶= sup τ ,τ ∈S Z d ∖ Λ ∣ ∣ Λ ∣ ∑ v ∈ Λ ⟨ f v,i ( σ )⟩ τ Λ − ⟨ f v,i ( σ )⟩ τ Λ ∣ , and(6.4) Fluc τ Λ ,i ( η ) ∶= sup τ ∈S Z d ∖ Λ ∣ ∣ Λ ∣ ∑ v ∈ Λ ⟨ f v,i ( σ )⟩ τ Λ − ⟨ f v,i ( σ )⟩ τ Λ ∣ . Let us note that these quantities only depend on the value of the field inside the set Λ, andthat we have the inequalities(6.5) 0 ≤ Fluc Λ ( η ) ≤ m ∑ i = Fluc Λ ,i ( η ) , Fluc Λ ( η ) ≤ τ Λ ( η ) and Fluc Λ ,i ( η ) ≤ τ Λ ,i ( η ) . Additionally, by the pointwise bound ∣ f v ( σ )∣ ≤
1, the four quantities (6.1), (6.2), (6.3) and (6.4)are bounded by 2 for any realization of the random field η . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 15
Structure of the random field.
Given an integer i ∈ { , . . . , m } and a vertex v ∈ Z d , wedenote by η i and η v,i the i -th component of η and η v respectively.Given a bounded set Λ ⊆ Z d , we denote by η Λ = ( η Λ , , . . . , η Λ ,m ) the restriction of the field η to the set Λ. We use the decomposition η Λ ∶= ( ˆ η Λ , η ⊥ Λ ) withˆ η Λ ∶= ∣ Λ ∣ − ∑ v ∈ Λ η v and η ⊥ Λ ∶= η Λ − ˆ η Λ . More generally, given a map w ∶ Λ → R m , we use the decomposition η Λ ∶= ( ˆ η w, Λ , η ⊥ w, Λ ) with,for each i ∈ { , . . . , m } ,ˆ η w, Λ ,i ∶= ∑ v ∈ Λ w i ( v ) ∑ v ∈ Λ w i ( v ) η v,i and η ⊥ w, Λ ,i ∶= η Λ ,i − ˆ η w, Λ ,i w i . We set ˆ η w, Λ ,i = w i = η is assumed to be a standard m -dimensional Gaussian vector, all the randomfields listed above are Gaussian. For each i ∈ { , . . . , m } , the variance of ˆ η Λ ,i and η Λ ,w,i are equal ∣ Λ ∣ − and 1 / (∑ v ∈ Λ w i ( v ) ) respectively. Moreover the fields ˆ η Λ and η ⊥ Λ are independent, andthe fields ˆ η w, Λ and η ⊥ w, Λ are independent.Given a function F depending on the realization of η in the set Λ and an integer i ∈ { , . . . , m } ,we may abuse notation and write F ( ˆ η Λ , η ⊥ Λ ) , F ( ˆ η Λ ,i , ( ˆ η Λ ,j ) j ≠ i , η ⊥ Λ ) or F ( ˆ η w, Λ , η ⊥ w, Λ ) instead of F ( η ) when we want to emphasize the dependence of the map F on a specific variable.For i ∈ { , . . . , m } , we denote by ∂F∂ ˆ η Λ ,i ( η ) and ∂F∂ ˆ η w, Λ ,i ( η ) the partial derivatives of the map F with respect to the variables ˆ η w, Λ ,i and ˆ η w, Λ ,i respectively.6.3. Notation for conditional expectation.
Given two random variables F and X dependingon the random field η such that E [∣ F ∣] < ∞ , we denote by E [ F ∣ X ] the conditional expectationof F with respect to X . Given an event A , we denote by P ( A ∣ X ) ∶= E [ A ∣ X ] the conditionalprobability. The random variable X will frequently be the fields ˆ η Λ , η ⊥ Λ , ˆ η w, Λ , η ⊥ w, Λ , the restrictionof the field η to some set Λ, or a combination of these options.Since the random variable E [ F ∣ X ] depends only on the realization of X , we may write E [ F ∣ X ] ( X ) instead of E [ F ∣ X ] ( η ) when we wish to make the dependence on the randomfield explicit. Similarly, we may write P ( A ∣ X ) ( X ) instead of P ( A ∣ X ) ( η ) .6.4. The domain subadditivity property.
In this section, we state a domain subadditivityproperty satisfied by the quantity Fluc Λ ( η ) . The result is a direct consequence of the consistencyproperty (2.5) and is used frequently in the proofs of Sections 7 and 8. Proposition 6.1 (Domain subadditivity property) . Let β > , η ∶ Z d → R m . Let Λ ′ , . . . , Λ ′ N ofbe a collection of disjoint bounded subsets of Z d and define Λ ′ ∶= ∪ Nj = Λ ′ j . Then we have Fluc Λ ′ ( η ) ≤ N ∑ j = ∣ Λ ′ j ∣∣ Λ ′ ∣ Fluc Λ ′ j ( η ) , as well as, for any integer i ∈ { , . . . , m } , sup τ ∈S Z d ∖ Λ ′ ∑ v ∈ Λ ′ ⟨ f v,i ( σ )⟩ τ Λ ′ ≤ N ∑ j = sup τ ∈S Z d ∖ Λ ′ j ∑ v ∈ Λ ′ j ⟨ f v,i ( σ )⟩ τ Λ ′ j and inf τ ∈S Z d ∖ Λ ′ ∑ v ∈ Λ ′ ⟨ f v,i ( σ )⟩ τ Λ ′ ≥ N ∑ j = inf τ ∈S Z d ∖ Λ ′ j ∑ v ∈ Λ ′ j ⟨ f v,i ( σ )⟩ τ Λ ′ j . Free energy: definition and basic properties.
The finite-volume free energy of therandom system is defined below (and corresponds to the one introduced in (5.1)).
Definition 6.2 (Finite-volume free energy) . Given a bounded domain Λ ⊆ Z d , an inversetemperature β >
0, a boundary condition τ ∈ S Z d ∖ Λ , a realization of the random field η , wedefine the free energy by the formulaFE τ Λ ( η ) ∶= − β ∣ Λ ∣ ln Z η,λβ, Λ ,τ = − β ∣ Λ ∣ ln ∫ S Λ exp (− βH η,λ Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) , where the integral is computed over the set of configurations satisfying σ = τ in Z d ∖ Λ.We next collect without proof some basic properties of the finite-volume free energy.
Proposition 6.3 (Properties of the free energy) . For any bounded domain Λ ⊆ Z d , any inversetemperature β > , and any boundary condition τ ∈ S Z d ∖ Λ , the map FE τ Λ satisfies the properties: ● Concavity and regularity:
The map η ↦ FE τ Λ ( η ) is concave and satisfies for any pair offields η, η ′ , ∣ FE τ Λ ( η ) − FE τ Λ ( η ′ )∣ ≤ λ ∣ Λ ∣ ∑ v ∈ Λ ∣ η v − η ′ v ∣ . ● Derivative:
The mapping η ↦ FE τ Λ ( η ) is differentiable and satisfies, for any i ∈{ , . . . , m } , (6.6) ∂ FE τ Λ ∂ ˆ η Λ ,i ( η ) = − λ ∣ Λ ∣ ∑ v ∈ Λ ⟨ f v,i ( σ )⟩ τ Λ . More generally, for any map w ∶ Λ ↦ R m which is not identically , (6.7) ∂ FE τ Λ ∂ ˆ η w, Λ ,i ( η ) = − λ ∣ Λ ∣ ∑ v ∈ Λ w i ( v ) ⟨ f v,i ( σ )⟩ τ Λ . ● Finite energy:
For any boundary condition τ ∈ S Z d ∖ Λ , any realization of random field η , (6.8) ∣ FE τ Λ ( η ) − FE τ Λ ( η )∣ ≤ C ∣ ∂ Λ ∣∣ Λ ∣ + Cλ ∑ v ∈ Λdist ( v,∂ Λ )≤ R ∣ η v ∣ . The proof of these results is a direct consequence of the formula for the free energy stated inDefinition 6.2, the assumption ∣ f v ( σ )∣ ≤ f v has range R and the inequality (2.2).In the case of systems equipped with a continuous symmetry, we make the dependence in theparameter h explicit, and refer to the free energy using the notation(6.9) FE τ,h Λ ( η ) ∶= − β ∣ Λ ∣ ln Z η,λ,hβ, Λ ,τ = − β ∣ Λ ∣ ln ∫ S Λ exp (− βH η,λ,h Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) . For any realization of the random field η , and any boundary condition τ ∈ S ∂ Λ , the mapping h ↦ FE τ,h Λ ( η ) is concave, 1-Lipschitz, differentiable, and satisfies, for any i ∈ { , . . . , n } ,(6.10) ∂ FE τ,h Λ ( η ) ∂h i = − ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ . As it will be used in the proofs below, we record that, since the identity (6.10) is valid for anydeterministic field η and any boundary condition τ ∈ S ∂ Λ , it implies the following result: forany random (measurable) boundary condition η ↦ τ ( η ) ∈ S ∂ Λ , ∂∂h i E [ FE τ ( η ) ,h Λ ( η )] = − E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ ( η ) ,h Λ ] . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 17
Moreover, the continuous symmetry of the systems manifests in the following ways: usingthe η → − η invariance of the law of the random field, we see that, for any random boundarycondition η ↦ τ ( η ) ∈ S ∂ Λ ,(6.11) ∀ v ∈ Λ , E [⟨ σ v ⟩ τ ( η ) ,h Λ ] = − E [⟨ σ v ⟩ ̃ τ ( η ) , − h Λ ] , where we used the notation ̃ τ ∶ η ↦ − τ (− η ) ∈ S ∂ Λ . In the case of the periodic boundary condition,the expected value of the magnetization does not depend on the vertex v , i.e.,(6.12) ∀ v ∈ Λ , E [⟨ σ v ⟩ per ,h Λ ] = E [⟨ σ ⟩ per ,h Λ ] , and when the magnetic field h is equal to 0, its value is equal to 0,(6.13) ∀ v ∈ Λ , E [⟨ σ v ⟩ per , ] = . Convention for constants.
Throughout this article, the symbols c and C denote positiveconstants which may vary from line to line, with C increasing and c decreasing. These constantsmay depend only on strength of the random field λ , the parameter m , the constant C H , theradius R and the map Ψ.7. Proofs for two-dimensional disordered spin systems
In this section, we study the general spin systems presented in Section 2 and prove Theorem 1,Theorem 2 and Corollary 2.2.In Section 7.1, we introduce the notion of δ -stability for λ -Lipschitz convex function andquantify the Lebesgue and Gaussian measures of the δ -stability set (see Proposition 7.3 andCorollary 7.4 below).The next three sections are devoted to the proof of Theorem 1 following the outline ofSection 5: Section 7.2 contains the proof of the estimate (5.3), in Section 7.3, we implementthe Mandelbrot percolation argument and prove the inequality (5.5), and in Section 7.4 wecomplete the proof of Theorem 1 by upgrading the stochastic integrability of (5.5).Sections 7.6 and 7.7 are devoted to the proofs of Theorem 2 and Corollary 2.2, respectively.7.1. A notion of δ -stability for real-valued convex functions. The following statement isa general result about real-valued convex functions; it asserts that if two λ -Lipschitz continuous,convex and differentiable functions are close (in the L ∞ -norm), then their derivatives cannot betoo distant from each other on a set of large Lebesgue measure. We first introduce the followingset. Definition 7.1.
Fix λ >
0. For each λ -Lipschitz, convex function g ∶ R → R , and each parameter r >
0, we define the set N λ,r ( g ) ∶= { g ∶ R → R ∶ g is convex, λ -Lipschitz continuous, differentiableand satisfies sup t ∈ R ∣ g ( t ) − g ( t )∣ ≤ r } . We then define the δ -stability set of the function g as follows. Definition 7.2 ( δ -stability set) . For each triplet of parameters λ, δ, r >
0, and each function g ∶ R → R convex, λ -Lipschitz and differentiable, we define the setStab ( λ, δ, r, g ) ∶= { t ∈ R ∶ ∃ g ∈ N λ,r ( g ) such that ∣ g ′ ( t ) − g ′ ( t )∣ > δ } . The next proposition estimates the Lebesgue measure of the set Stab ( λ, δ, r, g ) . Figure 1.
The figure represents a function g ∶ R → R which is convex and λ -Lipschitz. Thearea in blue represents the cylinder where the functions of the set N λ,r ( g ) must lie. An exampleof function g ∈ N λ,r ( g ) is drawn in orange. The set Stab ( λ, δ, r, g ) is drawn in red. Proposition 7.3 ( δ -stability for λ -Lipschitz convex functions) . There exists a numericalconstant C such that, for each λ > , each function g ∶ R → R convex, λ -Lipschitz anddifferentiable, and each pair of parameters r, δ > , (7.1) Leb ( Stab ( λ, δ, r, g )) ≤ Cλρδ . Proof.
We fix three parameters λ, δ, r >
0, a λ -Lipschitz, convex and differentiable function g ∶ R → R , and observe that if a point t ∈ R belongs to the set Stab ( λ, δ, r, g ) then there exists afunction g ∈ N λ,r ( g ) (which may depend on the value of t ), such that either:(1) The inequality g ′ ( t ) − g ′ ( t ) > δ holds.(2) Or the inequality g ′ ( t ) − g ′ ( t ) > δ holds.Let us first assume that the inequality (1) is satisfied; we claim that it implies the estimate(7.2) g ′ ( t + rδ ) ≥ g ′ ( t ) + δ . To prove (7.2), note that the assumption sup t ∈ R ∣ g ( t ) − g ( t )∣ ≤ r implies, for any s ∈ R ,(7.3) g ( s ) − r ≤ g ( s ) ≤ g ( s ) + r. Using the inequality g ′ ( t ) − g ′ ( t ) > δ and the convexity of the map g , we see that, for any s > t ,(7.4) g ( s ) ≥ g ( t ) + g ′ ( s )( s − t ) ≥ g ( t ) + ( g ′ ( t ) + δ ) ( s − t ) ≥ g ( t ) − r + ( g ′ ( t ) + δ ) ( s − t ) . A combination of the estimates (7.3) and (7.4) yields g ( s ) − g ( t ) s − t > g ′ ( t ) + δ − rs − t . Choosing the value s = t + r / δ in the previous inequality and using the convexity of g shows g ′ ( t + rδ ) ≥ g ( t + rδ ) − g ( t ) r / δ > g ′ ( t ) + δ − δ ≥ g ′ ( t ) + δ . The proof of the claim (7.2) is complete. In the case when the inequality (2) is satisfied, asimilar argument yields the estimate(7.5) g ′ ( t − rδ ) ≤ g ′ ( t ) − δ . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 19
A combination of (7.2) and (7.5), and the assumption that g is convex (which implies that itsderivative is increasing) shows that, for any point t ∈ Stab ( λ, δ, r, g ) ,(7.6) g ′ ( t + rδ ) ≥ g ′ ( t − rδ ) + δ . Using that the map g is convex and λ -Lipschitz, we see that, for any triplet of real numbers t − , t, t + ∈ R satisfying t − < t < t + ,(7.7) − λ ≤ g ′ ( t − ) ≤ g ′ ( t ) ≤ g ′ ( t + ) ≤ λ. The estimates (7.6) and (7.7) imply that there cannot exist a family t , . . . , t ⌊ λδ ⌋+ of (⌊ λδ ⌋ + ) -points satisfying the following properties:(1) For any pair of distinct integers i, j ∈ { , . . . , ⌊ λ / δ ⌋ + } , one has ∣ t i − t j ∣ > rδ .(2) For any integer i ∈ { , . . . , ⌊ λ / δ ⌋ + } , the point t i belongs to the set Stab ( λ, δ, r, g ) .This property implies that the set Stab ( λ, δ, r, g ) is included in the union of (at most) ⌊ λδ ⌋ intervals of length 16 r / δ which implies the upper boundLeb ( Stab ( λ, δ, r, g )) ≤ Cλrδ . This is (7.1). The proof of Proposition 7.3 is complete. (cid:3)
Proposition 7.3 implies a lower bound on the Gaussian measure of the set Stab ( λ, δ, r, g ) . Corollary 7.4.
There exists a numerical constant C > such that for each λ > , each λ -Lipschitz, convex and differentiable function g ∶ R → R , each pair of parameters δ, r > , andeach variance σ > such that r ≥ σδ / λ , √ πσ ∫ Stab ( λ,δ,r,g ) e − t σ dt ≤ − e − C λ r σ δ . Proof.
The proof is a consequence of the following inequality: there exists a numerical con-stant C > σ ∈ ( , ) , any real number α ≥ σ ,(7.8) sup A ⊆ R , Leb ( A )≤ α √ πσ ∫ A e − t σ dt = √ πσ ∫ α − α e − t σ dt ≤ − e − Cα σ . Corollary 7.4 is then obtained by combining the inequality (7.8) with the value α = Cλr / δ andthe estimate (7.1). (cid:3) A lower bound for the effect for the boundary condition on the averaged mag-netization.
In this section, we apply the result of Proposition 7.3 to the free energy associatedwith a discrete spin system and obtain the inequality (5.3). As it will be useful in the restof the proof of Theorem 1, we establish the result both in the case of boxes (see (7.9)) andannuli (see (7.10)). The lower bound is stated in the following lemma, and we recall thenotations Fluc Λ ( η ) , Fluc τ Λ ( η ) , Fluc Λ ,i ( η ) and Fluc τ Λ ,i ( η ) introduced in Section 6.1, as well asthe notation R for the range of the observables ( f v ) . Lemma 7.5.
Fix β > , λ > , and L ≥ . There exist two positive constants c, C ∈ ( , ∞) depending on the parameters λ , C H and R such that, for any δ > , (7.9) P ( Fluc Λ L ( η ) < δ + CL ) > exp (− Cδ ) . Additionally, for each vertex v ∈ Λ L and each integer L ′ ≤ L such that v + Λ L ′ ⊆ Λ L , (7.10) P ( Fluc Λ L ∖( v + Λ L ′ ) ( η ) < δ + CL ) > exp (− Cδ ) . Remark 7.6.
Before giving the proof, we mention that, under the additional assumption thatthe Hamiltonians ( H Λ ) satisfy the upper bounds ∣ H Λ ∣ ≤ C ∣ Λ ∣ (as is the case for the modelspresented in Section 2.2), the free energy is a locally Lipschitz continuous function of the inversetemperature β , and thus, a small perturbation of the parameter β only slightly modifies thefree energy. Consequently, the argument given below (which relies on the notion of δ -stabilityintroduced in Section 7.1) can be extended so that the result of Lemma 7.5 holds uniformlyover the parameter β when this quantity belongs to a small open set. Proof.
The argument relies on Proposition 7.3 and Corollary 7.4. Using the upper boundFluc Λ L ( η ) ≤
2, we may assume that δ ≤ i ∈ { , . . . , m } and anyfixed boundary condition τ ∈ S Z d ∖ Λ L ,(7.11) P ( Fluc τ Λ L ,i ( η ) < δ + CL ) > exp (− Cδ ) . We now fix an integer i ∈ { , . . . , m } and prove (7.11). We assume, without loss of generality,that L ≥ R .We introduce the notation(7.12) Fluc τ Λ L ,i ∶= sup τ ∈S Z d ∖ Λ L RRRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ ( L − R ) ⟨ f v,i ( σ )⟩ τ Λ − ⟨ f v,i ( σ )⟩ τ Λ RRRRRRRRRRRR , and make a few observations pertaining to this quantity. First, we have the volume estimates(7.13) ∣ Λ ( L − R ) ∣ ≥ c ∣ Λ L ∣ and ∣ Λ L ∖ Λ ( L − R ) ∣ ≤ C ∣ Λ L ∣ / L. Additionally, if we decompose the field η Λ L according to η Λ L ∶= ( ˆ η Λ L ,w , η ⊥ Λ L ,w ) , where w is theweight function given by w ∶= Λ L − R , then we have(7.14) ∂ FE τ Λ ∂ ˆ η w, Λ ,i ( η ) = − λ ∣ Λ L ∣ ∑ v ∈ Λ ( L − R ) ⟨ f v,i ( σ )⟩ τ Λ , and the random variable ˆ η Λ L ,w is independent of the realization of the random field η in theboundary layer Λ L ∖ Λ ( L − R ) . Moreover, using that the maps ( f v ) are bounded by 1, we have(7.15) ∣ Fluc τ Λ L ,i − Fluc τ Λ L ,i ∣ ≤ CL P − almost-surely . We then claim that the inequality (7.9) is implied by the conditional inequality(7.16) P ( Fluc τ Λ L ,i ( η ) < δ ∣ ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w ) > exp ⎛⎜⎝− Cδ ⎛⎝ + λL ∣ Λ L ∣ ∑ v ∈ Λ L ∖ Λ ( L − R ) ∣ η v ∣⎞⎠ ⎞⎟⎠ P − a.s.Indeed, taking the expectation in (7.16) shows P ( Fluc τ Λ L ,i ( η ) < δ ) = E [ P ( Fluc τ Λ L ,i ( η ) < δ ∣ ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w )] (7.17) > E ⎡⎢⎢⎢⎢⎢⎣ exp ⎛⎜⎝− Cδ ⎛⎝ + Lλ ∣ Λ L ∣ ∑ v ∈ Λ L ∖ Λ ( L − R ) ∣ η v ∣⎞⎠ ⎞⎟⎠⎤⎥⎥⎥⎥⎥⎦> exp (− Cδ ) . where we used that the random variables ( η v ) v ∈ Λ L ∖ Λ ( L − R ) are Gaussian, independent, and thesecond volume estimate stated in (7.13). Combining (7.17) with the pointwise bound (7.15)completes the proof of (7.9). UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 21
We now focus on the proof of (7.16). To this end, let us fix a realization of the averagedfields ( ˆ η Λ L ,w,j ) j ≠ i and of the orthogonal field η ⊥ Λ L ,w . We define(7.18) g ∶ ˆ η Λ L ,w,i ↦ − FE τ Λ L ( ˆ η Λ L ,w,i , ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w ) and r ∶= C ∣ ∂ Λ L ∣∣ Λ L ∣ + Cλ ∣ Λ L ∣ ∑ v ∈ Λ L ∖ Λ ( L − R ) ∣ η v ∣ , where C is the constant appearing the the right-hand side of (6.8). We note that, by Proposi-tion 6.3, the function g is convex, λ -Lipschitz, differentiable and its derivative is given by theformula (7.14).The estimate (6.8) can be rewritten as follows: for any boundary condition τ ∈ S Z d ∖ Λ L , themap ˆ η Λ L ,w,i ↦ − FE τ Λ L ( ˆ η Λ L ,w,i , ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w ) belongs to the space N λ,r ( g ) . The previous observation combined with the identity (6.6) yields the inclusion of sets { ˆ η Λ L ,w,i ∈ R ∶ Fluc τ Λ L ,w,i ( ˆ η Λ L ,w,i , ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w ) ≥ δ } ⊆ Stab ( λ, δ, r, g ) . Applying Proposition 7.3 and the formula for the parameter r stated in (7.18), we deduceLeb ({ ˆ η Λ L ,w,i ∈ R ∶ Fluc τ Λ L ,i ( ˆ η Λ L ,w,i , ( ˆ η Λ L ,w,j ) j ≠ i , η ⊥ Λ L ,w ) ≥ δ }) (7.19) ≤ C ∣ ∂ Λ L ∣ δ ∣ Λ L ∣ + Cδ ∣ Λ L ∣ ∑ v ∈ Λ L ∖ Λ ( L − R ) ∣ η v ∣≤ Cδ L + Cδ ∣ Λ L ∣ ∑ v ∈ Λ L ∖ Λ ( L − R ) ∣ η v ∣ , where we used the upper bound ∣ ∂ Λ L ∣ / ∣ Λ L ∣ ≤ C / L in the second inequality. Using that the ran-dom variable ˆ η Λ L ,w,i is Gaussian of variance ∣ Λ ( L − R ) ∣ − ≥ cL − , that the random variables ˆ η Λ L ,w,i , ( ˆ η Λ L ,w,j ) j ≠ i and η ⊥ Λ L ,w are independent, and Corollary 7.4, we obtain that the inequality (7.19)implies the estimate (7.16). The proof of (7.9) is complete.The proof of the inequality (7.10) only requires a notational modification of the previousargument, we thus omit the details. (cid:3) Mandelbrot percolation argument.
In this section, we combine the result obtained inLemma 7.5 with a Mandelbrot percolation argument to obtain the quantitative estimate (5.5).The result is stated in the following lemma.
Lemma 7.7.
Fix L ≥ , β > , λ > . There exist two positive constants c, C ∈ ( , ∞) such that (7.20) P ( Fluc Λ L ( η ) < C √ ln ln L ) ≥ − exp (− c √ ln L ) . Proof.
We set δ ∶= C / √ ln ln L ∧
4, for some large constant C ≥ √ ln ln L ≤ C √ L , for any L ≥ C is large enough so that: for any box Λ ⊆ Z d of side length (cid:96) ≥ √ L , and any box Λ ′ ⊆ Λ whose side length is smaller than (cid:96) / P ( Fluc Λ ∖ Λ ′ < δ ) > exp (− Cδ ) . The strategy is to implement a Mandelbrot percolation argument. To this end, we define thefollowing notion of good box: a box Λ ′ ⊆ Λ L is good if and only if(7.22) Fluc Λ ′ ( η ) ≤ δ. We say that a box is bad if it is not good. Let us recall that the event (7.22) only depends onthe realization of the random field η inside the box Λ ′ (since the random variable Fluc Λ ′ onlydepends on the value of the field η inside the box Λ ′ ) Figure 2.
A realization of the Mandelbrot percolation with the values k = l max =
3. Thebad cubes are drawn in black.
Let us first introduce a few additional notations. Given an integer k ≥
2, we denote by l max the largest integer which satisfies k l max ≤ √ L , i.e., l max = ⌊ ln L /( k )⌋ . For each integer l ∈ { , . . . , l max } , we introduce the set of boxes(7.23) T l ∶= {( z + [− Lk l , Lk l ) d ) ∩ Z d ∶ z ∈ Lk l Z d ∩ Λ L } . We note that that, for each integer l ∈ { , . . . , l max } , the collection of boxes T l forms a partitionof the box Λ L . Additionally, two boxes of the collection ⋃ l max l = T l are either disjoint or includedin one another. For each vertex v ∈ Λ L , and each integer l ∈ { , . . . , l max } , we denote by Λ l ( v ) the unique box of the set T l containing the point v .We select the integer k to be the smallest integer larger or equal to 2 such that the followingproperties are satisfied: l max ≥ ∀ v ∈ Λ L , ∀ l ∈ { , . . . , l max − } , ∣ Λ l + ( v )∣ ≤ δ ∣ Λ l ( v )∣ We remark that this integer always exists if the constant C is chosen large enough, and thatthere exist two (numerical) constants C, c such that cδ − ≤ k ≤ Cδ − .We then construct recursively a (random) sequence of collection of good boxes Q l ⊆ T l , for l ∈ { , . . . , l max } , according to the following algorithm: ● Initiation: we set Q = { Λ L } if Λ is a good box and Q = ∅ otherwise; ● Induction step: we assume that the sets Q , . . . , Q l − have been constructed, and wishto construct the collection Q l . We consider the set of boxes T l to which we remove allthe boxes which are included in a box of the collection ⋃ l − i = Q i , that is, we define the set T ′ l ∶= { Λ ′ ∈ T l ∶ ∀ Λ ′′ ∈ l − ⋃ i = Q i , Λ ′ /⊆ Λ ′′ } . We define the set Q l to be the set of boxes which are good and belong to T ′ l , i.e., Q l ∶= { Λ ′ ∈ T ′ l ∶ Λ ′ is good } . We then define
Q ∶= ∪ l max l = Q l . Let us note that two boxes in the set Q are either equal ordisjoint. The collection of boxes Q is not in general a partition of the box Λ L , and there is anon-empty set of uncovered points which can be characterized by the following criterion:(7.24) v ∈ Λ L is uncovered ⇐⇒ ∀ l ∈ { , . . . , l max } , Λ l ( v ) is a bad box . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 23
We next show that the set of uncovered points is small. To this end, we show the followingupper bound on the probability of a vertex v ∈ Λ L to be uncovered: there exists a constant c such that(7.25) P ( v is uncovered ) ≤ exp (− c √ ln L ) . To prove the inequality (7.25), we fix a vertex v ∈ Λ L , and rewrite the equivalence (7.24) asfollows:(7.26) { v is uncovered } = l max ⋂ l = { Fluc Λ l ( v ) > δ } . The strategy is then to prove that the l max events in the right side of (7.26) are well-approximatedby independent events, and to use the independence in order to estimate the probability of theirintersection. By the domain subadditivity property stated in Proposition 6.1, and the pointwisebound Fluc Λ ≤
2, we have, for any v ∈ Λ L and any l ∈ { , . . . , l max − } ,Fluc Λ l ( v ) ≤ ∣ Λ l ( v ) ∖ Λ l + ( v )∣∣ Λ l ( v )∣ Fluc Λ l ( v )∖ Λ l + ( v ) ( η ) + ∣ Λ l + ( v )∣∣ Λ l ( v )∣ Fluc Λ l + ( v ) (7.27) ≤ Fluc Λ l ( v )∖ Λ l + ( v ) ( η ) + ∣ Λ l + ( v )∣∣ Λ l ( v )∣≤ Fluc Λ l ( v )∖ Λ l + ( v ) ( η ) + δ , where we used in the last inequality that, by the definition of the integer k , the ratio of thevolumes of the boxes Λ l + ( v ) and Λ l ( v ) is smaller than 4 / δ . The estimate (7.27) implies theinclusion of events(7.28) { v is uncovered } ⊆ l max − ⋂ l = { Fluc Λ l ( v )∖ Λ l + ( v ) ( η ) ≥ δ } Using that the annuli ( Λ l ( v ) ∖ Λ l + ( v )) l ∈{ ,...,l max − } are disjoint and that the random variablesFluc Λ l ( v )∖ Λ l + ( v ) ( η ) depend only on the restriction of the random field to the annulus Λ l ( v ) ∖ Λ l + ( v ) , we obtain that the events in the right side of (7.28) are independent. We deduce that P ( v is uncovered ) ≤ l max − ∏ l = P ( Fluc Λ l ( v )∖ Λ l + ( v ) ( η ) ≥ δ ) . We recall the definition of the parameter δ and of the integer l max stated at the beginning ofthe proof. Using (7.21), we obtain that P ( v is uncovered ) ≤ ( − exp (− Cδ )) l max ≤ ⎛⎝ − ( ln L ) C / C ⎞⎠ c ln L ln ln ln L . Choosing the constant C large enough, e.g. larger than √ C , we obtain P ( v is uncovered ) ≤ exp (− c √ ln L ) . The proof of (7.25) is complete. We now use the inequality (7.25) to complete the proof of theestimate (7.20). By the domain subadditivity property for the quantity Fluc Λ and the pointwise bound the pointwise bound Fluc Λ ≤
2, we haveFluc Λ L ( η ) ≤ ∑ Λ ∈Q ∣ Λ ∣∣ Λ L ∣ Fluc Λ ( η ) + ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣ (7.29) ≤ ∑ Λ ∈Q ∣ Λ ∣∣ Λ L ∣ δ + ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣≤ δ + ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣ . By using (7.25) and Markov’s inequality, we have P ( ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣ ≥ δ ) ≤ E [ ∣ Λ L ∣ ∑ v ∈ Λ L { v is uncovered } ] δ ≤ ∑ v ∈ Λ L P ( v is uncovered )∣ Λ L ∣ δ (7.30) ≤ exp (− c √ ln L ) δ ≤ exp (− c √ ln L ) , by reducing the value of the constant c in the last inequality. We have thus obtained P ( Fluc Λ L ( η ) > δ ) ≤ exp (− c √ ln L ) . This is (7.20). The proof of Lemma 7.7 is complete. (cid:3)
Upgrading the stochastic integrability.
This section is the final step of the proof ofTheorem 1. We use a concentration argument combined with the domain subadditivity propertyapplied to the quantity Fluc Λ to upgrade the stochastic integrability obtained in Lemma 7.7. Proof of (2.6) of Theorem 1.
We split the box Λ L into (approximately) N ≃ L boxes of sidelength of order L . We denote these boxes by ̃ Λ , . . . , ̃ Λ N . By the domain subadditivity propertyfor the quantity Fluc h Λ , we have the inequality(7.31) Fluc Λ L ( η ) ≤ N N ∑ i = Fluc ̃ Λ i ( η ) . Since the boxes ̃ Λ , . . . , ̃ Λ N are disjoint, the sum in the right side is a sum of independentrandom variables, to which we can apply a concentration argument. To implement this strategy,we fix an integer i ∈ { , . . . , N } , apply the inequality (7.20) to the box ̃ Λ i (the result was provedfor the boxes of the form Λ L for L ≥ Z d by translationinvariance of the random field), together with the bound Fluc ̃ Λ i ≤
2, and the fact that that theside length of the box ̃ Λ i is larger than cL . We obtain, for any integer i ∈ { , . . . , N } ,(7.32) E [ Fluc ̃ Λ i ] ≤ C √ ln ln L / ≤ C √ ln ln L .
We use the inequalities (7.31), (7.32) to obtain, for any K ≥ P [ Fluc Λ L ≥ C √ ln ln L + K ] ≤ P [ N N ∑ i = ( Fluc ̃ Λ i − E [ Fluc ̃ Λ i ]) ≥ K ] . Using that the random variables Fluc ̃ Λ , . . . , Fluc ̃ Λ N are i.i.d., non-negative, bounded by 2almost surely together with Hoeffding’s inequality, we obtain, for any K ≥ P [ Fluc Λ L ≥ C √ ln ln L + K ] ≤ Ce − cNK . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 25
Recalling that N is comparable to L , and choosing K = C √ ln ln L , we have obtained P [ Fluc Λ L ≥ C √ ln ln L ] ≤ C exp (− cL / √ ln ln L ) ≤ C exp (− cL ) . This is the estimate (2.6). The proof is complete. (cid:3)
The translation-invariant setup.
In this section, we prove the estimate (2.7) pertainingto the translation invariant setup and thus complete the proof of Theorem 1
Proof of (2.7) of Theorem 1.
We assume in this proof that the model is translation invariantand satisfies the corresponding additional assumptions stated in Section 2.1. We fix an integer i ∈ { , . . . , m } .First, using the upper bound ∣ f v ∣ ≤ L ∈ N , E ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ − E ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ (7.33) ≤ E ⎡⎢⎢⎢⎢⎣ sup τ ,τ ∈S Z d ∖ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f (T v σ )⟩ τ Λ L − ⟨ f (T v σ )⟩ τ Λ L RRRRRRRRRRR⎤⎥⎥⎥⎥⎦≤ C √ ln ln L .
In the translation-invariant setup, we may combine the inequalities stated in Proposition 6.1with a subadditivity argument to obtain the following convergences(7.34) E ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ —→ L →∞ inf L ∈ N E ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ , E ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ —→ L →∞ sup L ∈ N E ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ . Using the inequality (7.33), we see that the two limits in the right-hand sides of (7.34) areequal. In particular, we may define α i ∶= inf L ∈ N E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ sup τ ∈S Z d ∖ Λ L ∑ v ∈ Λ ′ ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ = sup L ∈ N E [ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ ′ ⟨ f ,i (T v σ )⟩ τ Λ L ] . Additionally, we see that, for any integer L ∈ N ,(7.35) E ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ ≤ α i ≤ E ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ A combination of (7.33) and (7.34) thus yields
RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ − α i RRRRRRRRRRRR ≤ C √ ln ln L ,
RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L ⎤⎥⎥⎥⎥⎦ − α i RRRRRRRRRRRR ≤ C √ ln ln L .
Consequently, using the inequalities stated in Proposition 6.1 together with the same concentra-tion argument as the one developed in the proof of Lemma 7.7, we obtain inequalities(7.36) P ⎡⎢⎢⎢⎢⎣ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L − α i ≥ C √ ln ln L ⎤⎥⎥⎥⎥⎦ ≤ exp (− cL ) , P ⎡⎢⎢⎢⎢⎣ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L − α i ≤ − C √ ln ln L ⎤⎥⎥⎥⎥⎦ ≤ exp (− cL ) . We then note that the following inclusion of events holds(7.37) ⎧⎪⎪⎨⎪⎪⎩ sup τ ∈S Z d ∖ Λ L RRRRRRRRRRR α i − ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L RRRRRRRRRRR ≥ C √ ln ln L ⎫⎪⎪⎬⎪⎪⎭⊆ ⎧⎪⎪⎨⎪⎪⎩ sup τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L − α i ≥ C √ ln ln L ⎫⎪⎪⎬⎪⎪⎭⋃ ⎧⎪⎪⎨⎪⎪⎩ inf τ ∈S Z d ∖ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L − α i ≤ − C √ ln ln L ⎫⎪⎪⎬⎪⎪⎭ . A combination of (7.36), (7.37) and a union bound then yields(7.38) P ⎛⎝ sup τ ∈S Z d ∖ Λ L RRRRRRRRRRR α i − ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ f ,i (T v σ )⟩ τ Λ L RRRRRRRRRRR > C √ ln ln L ⎞⎠ ≤ exp (− cL ) . Since the inequality (7.38) is valid for any integer i ∈ { , . . . , m } , we obtain the inequality (2.7)of Theorem 1 with the value α ∶= ( α , . . . , α m ) . (cid:3) Proof of Theorem 2.
The objective of this section is to generalize Theorem 1. We provethat for any L ≥
4, any box Λ ⊆ Z d of side length L , and any deterministic weight function w ∶ Λ → [− , ] m , the quantityFluc Λ ,w ( η ) ∶= sup τ ,τ ∈S Z d ∖ Λ ∣ Λ ∣ ∣ ∑ v ∈ Λ w ( v ) ⟨ f v ( σ )⟩ τ Λ − ⟨ f v ( σ )⟩ τ Λ ∣ is smaller than √ ln ln L with high probability. Theorem 1 corresponds to the case when thefunction w is constant equal to 1. Let us note that the quantity Fluc Λ ,w ( η ) depends only onthe realization of the field η inside the box Λ, satisfies the same domain subadditivity propertyas the one stated in Proposition 6.1, and that, by the assumption ∣ w ( v )∣ ≤
1, one has the bound0 ≤ Fluc Λ ,w ( η ) ≤ η .The argument is similar to the one developed in the proof of Theorem 1, the main differenceis that we rely on the identity (6.7) to obtain information on the observable Fluc Λ L ,w ( η ) insteadof (6.6). The fact that the map w can take small values must be taken into account in theargument and causes a slight deterioration of the rate of convergence: we obtain the quantitativerate √ ln ln L instead of the rate √ ln ln L obtained in Theorem 1. Proof of Theorem 2.
The strategy of the argument is similar to the proof of Theorem 1. Weonly present a detailed sketch of the argument pointing out the main differences with the proofof Theorem 1. The first step is to prove the estimate: for any box Λ ⊆ Λ L of side length largerthan √ L such that ∑ v ∈ Λ ∣ w ( v )∣ ≥ ∣ Λ ∣ / √ ln ln L , and any δ > P ( Fluc Λ ,w ( η ) < δ ) ≥ exp (− C √ ln ln Lδ ) . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 27
The proof is similar to the the proof of Lemma 7.5; the main differences are that we needto decompose the random field η according to the formula, for each i ∈ { , . . . , m } , η ∶=( η w, Λ ,i , ( η w, Λ ,j ) j ≠ i , η ⊥ w, Λ ) (following the notation introduced in Section 6.2), use the identity (6.7)(instead of (6.6)), and use Corollary 7.4 with the variance σ ∶= ∑ v ∈ Λ w i ( v ) / ∣ Λ ∣ instead of σ = / ∣ Λ ∣ .The second step of the argument corresponds to Section 7.3. We combine the inequality (7.39)with a Mandelbrot percolation argument and prove the estimate(7.40) P ( Fluc Λ ,w ( η ) < C √ ln ln L ) ≥ − exp (− c √ ln L ) . To this end, we set δ ∶= C / √ ln ln L ∨
4, for some large constant C ∈ [√ , ∞) , and define thefollowing notion of good box: a box Λ ⊆ Λ L is said to be good if and only if(7.41) Fluc Λ ,w ( η ) ≤ δ or 1 ∣ Λ ∣ ∑ v ∈ Λ w ( v ) ≤ √ ln ln L .
Let us note that, for any box Λ ⊆ Λ L , the assumption ∣ f ( σ )∣ ≤ Λ ,w ( η ) ≤ ( ∣ Λ ∣ ∑ v ∈ Λ ∣ w ( v )∣ ) ⎛⎝ sup τ ,τ ∈S Z d ∖ Λ ∣ Λ ∣ ∑ v ∈ Λ ∣⟨ f v ( σ )⟩ τ Λ − ⟨ f v ( σ )⟩ τ Λ ∣ ⎞⎠ ≤ √ ¿``(cid:192) ∣ Λ ∣ ∑ v ∈ Λ ∣ w ( v )∣ . Consequently, if a box Λ is good, then we have Fluc Λ ,w ( η ) ≤ δ (under the nonrestrictiveassumption C ≥ √ Q of good boxes such that the set of uncovered points is small: we obtain theinequality(7.42) P ( v is uncovered ) ≤ exp (− c √ ln L ) . We now use the estimate (7.42) to prove the inequality (7.40). Using the domain subadditivityproperty for the quantity Fluc Λ ,w and the upper bound Fluc Λ ,w ≤
2, we haveFluc Λ L ,w ( η ) ≤ ∑ Λ ∈Q ∣ Λ ∣∣ Λ L ∣ Fluc Λ ,w ( η ) + ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣ (7.43) ≤ δ + ∣ Λ L ∖ ⋃ Λ ∈Q Λ ∣∣ Λ L ∣ . We then estimate the second term in the right side of (7.43) by combining Markov’s inequalitywith the estimate (7.42) as was done in the computation (7.30). The concentration argument isessentially identical to the one presented in the proof of Theorem 1, we thus omit the details. (cid:3)
Proof of Corollary 2.2.
As a corollary of Theorem 2, we show that the absolute valueof the expectation of the thermal expectations ⟨ f v ( σ )⟩ τ Λ L and ⟨ f v ( σ )⟩ τ Λ L is quantitatively smallfor any pair of random boundary conditions η ↦ τ ( η ) , τ ( η ) . Proof of Corollary 2.2.
We select a pair of random (measurable) boundary conditions η ↦ τ ( η ) , τ ( η ) ∈ S Z d ∖ Λ L , and define the deterministic weight function w according to the formula(7.44) ∀ v ∈ Λ L , w ( v ) ∶= ⎧⎪⎪⎪⎨⎪⎪⎪⎩ E [⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L ] ≥ , − E [⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L ] < . Applying Theorem 2 yields(7.45) E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L w ( v ) (⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L )⎤⎥⎥⎥⎥⎦ ≤ E [ Fluc Λ L ,w ( η )] ≤ C √ ln ln L .
We can then estimate the left-hand side of (7.45). We obtain E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L w ( v ) (⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L )⎤⎥⎥⎥⎥⎦ = ∣ Λ L ∣ ∑ v ∈ Λ L w ( v ) E [⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L ]= ∣ Λ L ∣ ∑ v ∈ Λ L ∣ E [⟨ f v ( σ )⟩ τ ( η ) Λ L − ⟨ f v ( σ )⟩ τ ( η ) Λ L ]∣ . A combination of the two previous displays completes the proof of Corollary 2.2. (cid:3) Proofs for spin systems with continuous symmetry
In this section, we study the spin systems with continuous symmetry presented in Section 3and prove Theorem 3 and Theorem 4. The section is organized as follows: ● In Section 8.1, we establish two variational lemmas on the set of bounded functions whoseintegral on any interval of the real line R is bounded. ● In Section 8.2, we implement a Mermin-Wagner type argument to prove an upper bound onthe free energy (see Proposition 8.4). ● Section 8.3 is devoted to the proof of Theorem 3 in the subcritical dimensions d = , ,
3. InSubsection 8.3.1, we combine the Mermin-Wagner upper bound obtained in Proposition 8.4with Lemma 8.1 to prove the algebraic decay of the thermally and spatially averaged magne-tization with fixed boundary condition stated in (3.4). In Subsection 8.3.1, we build uponthe results of Subsection 8.3.1 and prove the inequality (3.5), thus completing the proof ofTheorem 3. ● Section 8.4 is devoted to the proof of Theorem 4 following the outline of Section 5 and isdivided into three subsections. In Subsection 8.6, we combine the Mermin Wagner upperbound of Proposition 8.4 and Lemma 8.2 and establish that, given a box Λ ⊆ Λ L , if theaveraged field ˆ η Λ ,i is negative enough, then the thermally and spatially averaged magnetizationmust be small (see Lemma 8.6). In Subsection 8.3.1, we combine the result of Lemma 8.1with a Mandelbrot percolation argument and obtain the quantitative estimate stated inLemma 8.7 on the expectation of the spatially and thermally averaged magnetization with afixed boundary condition. Finally in Subsection 8.4.3, we upgrade the result of Lemma 8.7to include a supremum over all the possible boundary conditions, and complete the proof ofTheorem 4.8.1. Variational lemmas.
In this section, we state and prove two variational lemmas onthe set of measurable bounded functions defined on R and whose integral on every interval isbounded in absolute value by 1, i.e.,(8.1) G ∶= { g ∶ R → R ∶ g is measurable, bounded and, for any real interval I ⊆ R , ∣∫ I g ( t ) dt ∣ ≤ } . The first result we establish asserts that the Gaussian expectation of any map g ∈ G isbounded by an explicit constant. Proposition 8.1.
One has the identity (8.2) sup g ∈G ∣∫ R g ( t ) e − t dt ∣ ≤ ∫ ∞ te − t dt. UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 29
Proof.
We fix a function g ∈ G , and let G ( t ) ∶= ∫ t g ( s ) ds . By the definition of the set G , themap G satisfies ∣ G ( t )∣ ≤ t ∈ R , the identity G ( ) =
0, and is Lipschitz continuous. Byperforming an integration by parts, we obtain ∣∫ R g ( t ) e − t dt ∣ = ∣∫ R G ( t ) te − t dt ∣ ≤ ∫ R ∣ t ∣ e − t dt = ∫ ∞ te − t dt. (cid:3) The next lemma provides a lower bound on the Gaussian measure of the set { g ≤ δ } , for any g ∈ G satisfying g ≥ −
1, and any δ > Lemma 8.2.
There exists a numerical constant C > such that, for any δ ∈ ( , ] , (8.3) inf g ∈G g ≥− ∫ R { g ( t )≤ δ } e − t / dt ≥ e − C / δ . Proof.
We select a function g ∈ G and let G , G ∶ R → R be the maps defined by the formulas ∀ t ∈ R , G ( t ) ∶= ∫ t { g ( s )≤ δ } ds and G ( t ) ∶= ∫ t − t { g ( s )≤ δ } ds. Note that the functions G and G are increasing, 1 and 2-Lipschitz continuous respectively, andsatisfy the identity G ( t ) = G ( t ) − G (− t ) . By performing an integration by parts and a changeof variable, we see that ∫ R { g ( t )< δ } e − t dt = ∫ R G ( t ) te − t dt = ∫ ∞ ( G ( t ) − G (− t )) te − t dt (8.4) = ∫ ∞ G ( t ) te − t dt. We next claim that the map G satisfies the lower bound(8.5) ∀ t ≥ , G ( t ) ≥ max ( , δt − + δ ) . To prove (8.5), we use the assumption g ≥ − g ≥ δ { g > δ } − { g ≤ δ } ≥ δ ( − { g ≤ δ } ) − { g ≤ δ } ≥ δ − ( + δ ) { g ≤ δ } . Integrating this inequality over the interval [− t, t ] and using the properties on the function g ,we obtain(8.6) 1 ≥ δt − ( + δ ) G ( t ) ⇐⇒ G ( t ) ≥ δt − + δ . We conclude the proof of (8.5) by using that G is non-negative. A combination of (8.4) and (8.5)implies the inequality (cid:3) (8.7) inf g ∈G g ≥− ∫ R { g ( t )≤ δ } e − t dt ≥ ∫ ∞ δ δt − + δ te − t dt ≥ e − C / δ . A Mermin-Wagner upper bound for the free energy.
In this section, we obtainan upper bound on the free energy of a spin system equipped with a continuous symmetryby implementing a Mermin-Wagner argument. We recall that the spin space is assumed tobe the sphere S n − , for some n ≥
2, as well as the notation for conditional expectations andprobabilities introduced in Section 6.3 (as they will be used frequently in the proofs below).Before stating the result, we introduce the following definition.
Definition 8.3 (Free energy) . Let Λ , Λ be two boxes of Z d such that Λ ⊆ Λ . For any field η ∶ Λ → R , we denote by ̃ η Λ , Λ (8.8) ̃ η Λ , Λ ,v ∶= { η v if v ∈ Λ ∖ Λ , − η v if v ∈ Λ . We define the free energy, for any η ∶ Λ → R , ̃ FE τ,h Λ , Λ ( η ) ∶= FE τ,h Λ (̃ η Λ , Λ ) . The main result of this section is an upper bound on the difference of the free energies FE τ,h Λ and ̃ FE τ,h Λ , Λ conditionally on the values of the field in the box Λ and outside the box 2Λ. Proposition 8.4 (Mermin-Wagner upper bound for the energy) . Let n ≥ , d ∈ { , , , } and i ∈ { , . . . , n } . Let β > be the inverse temperature, λ > be the disorder strength and h ∈ R n be the deterministic external field. Fix a box Λ ⊆ Z d of side length L , and let τ ∈ S ∂ Λ be aboundary condition. For any box Λ of side length (cid:96) such that ⊆ Λ , we have the estimate (8.9) E [ ̃ FE τ,h Λ , Λ − FE τ,h Λ ∣ η ( Λ ∖ )∪ Λ ,i ] ≤ C(cid:96) d − L d + (cid:96) d L d ∣ h ∣ P − almost-surely , Proof.
Let us fix two boxes Λ , Λ satisfying 2Λ ⊆ Λ , of side lengths L and (cid:96) respectively, and aboundary condition τ ∈ S ∂ Λ . We recall the notations e , . . . , e n for the canonical basis of R n and Λ + ∶= Λ ∪ ∂ Λ . All the configurations σ ∈ S Λ + in this proof are implicitly assumed to satisfy σ ∂ Λ = τ .Let us consider a smooth map r ∶ R → O ( n ) satisfying r = r π = I n , for any pair θ , θ ∈ R , r θ ○ r θ = r θ + θ , and such that r π ( e i ) = − e i . For each vertex v ∈ Z d , we denote by(8.10) θ v ∶= ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ v ∈ Z d ∖ ,π ( dist ( v, ∂ ( )) (cid:96) ∨ ) if v ∈ . We then define two rotations R, ̃ R on the space of configurations by the formulas, for any σ ∈ S Λ + and any vertex v ∈ Z d , ( Rσ ) v = r θ v σ v and ( ̃ Rσ ) v = r − θ v σ v . We extend the domain of the rotations R, ̃ R to the set of fields, and write, for any realization ofthe random field η and any vertex v ∈ Λ , ( Rη ) v = r θ v η v and ( ̃ Rη ) v = r − θ v η v . For any realization of the field η and any configuration σ ∈ S Λ + , we have the identities(8.11) ∑ v ∈ Λ ( Rη ) v ⋅ ( Rσ ) v = ∑ v ∈ Λ η v ⋅ σ v and ∑ v ∈ Λ ( ̃ Rη ) v ⋅ ( ̃ Rσ ) v = ∑ v ∈ Λ η v ⋅ σ v . Additionally, since the two rotations R and ̃ R are equal outside the box 2Λ and inside Λ, wehave(8.12) RRRRRRRRRRR ∑ v ∈ Λ h ⋅ ( Rσ ) v − ∑ v ∈ Λ h ⋅ ( ̃ Rσ ) v RRRRRRRRRRR = ∣ ∑ v ∈ ∖ Λ h ⋅ ( Rσ ) v − ∑ v ∈ ∖ Λ h ⋅ ( ̃ Rσ ) v ∣ ≤ C(cid:96) d ∣ h ∣ We next prove the inequality, for any configuration σ ∈ S Λ + ,(8.13) ∑ v,w ∈ Λ + v ∼ w Ψ ( r θ v σ v , r θ w σ w ) + ∑ v,w ∈ Λ + v ∼ w Ψ ( r − θ v σ v , r − θ w σ w ) ≤ ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , σ w ) + C(cid:96) d − , UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 31 for some constant C depending only on the map Ψ and the rotation r . To prove the inequal-ity (8.13), we first use that the map Ψ is invariant under the rotations r θ v and r − θ v , and theproperties of the map r . We obtain(8.14) ∑ v,w ∈ Λ + v ∼ w Ψ ( r θ v σ v , r θ w σ w ) + ∑ v,w ∈ Λ + v ∼ w Ψ ( r − θ v σ v , r − θ w σ w )= ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ w − θ v σ w ) + ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ v − θ w σ w ) . Using that the map Ψ is assumed to be continuously twice differentiable and bounded, that themap r is smooth and that the state space S n − is compact, we can perform a Taylor expansionand obtain that there exists a constant C , depending on the maps Ψ and r such that, for each θ ∈ R , and each pair of spins σ , σ ∈ S n − ,(8.15) ∣ Ψ ( σ , r θ σ ) + Ψ ( σ , r − θ σ ) − ( σ , σ )∣ ≤ Cθ . Applying the inequality (8.15) with the values θ = θ v − θ w , σ = σ v , σ = σ w , and summing overall the pairs of neighboring vertices v, w in Λ + yields(8.16) ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ w − θ v σ w )+ ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ v − θ w σ w ) ≤ ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , σ w )+ C ∑ v,w ∈ Λ + v ∼ w ∣ θ v − θ w ∣ . By the definition of the map θ stated in (8.10), we have, for any pair of neighboring vertices v, w ∈ Λ + ,(8.17) ⎧⎪⎪⎪⎨⎪⎪⎪⎩∣ θ v − θ w ∣ = v, w ∈ Λ ∪ ( Λ ∖ ) , ∣ θ v − θ w ∣ ≤ C(cid:96) if { v, w } ∩ ( ∖ Λ ) ≠ ∅ . Combining the estimates (8.16), (8.17), and using that the volume of the annulus ( ∖ Λ ) is oforder (cid:96) d , we obtain(8.18) ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ w − θ v σ w ) + ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , r θ v − θ w σ w ) ≤ ∑ v,w ∈ Λ + v ∼ w Ψ ( σ v , σ w ) + C(cid:96) d − . Combining the identity (8.14) and the inequality (8.18) completes the proof of (8.13). Combiningthe estimates (8.11), (8.12) and (8.13) with the definition of the noised Hamiltonian (3.2), wehave obtained the inequality: for any realization of the random field η and any configuration σ ∈ S Λ + ,(8.19) H Rη,h Λ ( Rσ ) + H ̃ Rη,h Λ ( ̃ Rσ ) ≤ H η,h Λ ( σ ) + C(cid:96) d − + C(cid:96) d ∣ h ∣ . We now use the inequality (8.19) to prove the estimate (8.9). By the rotational invariance ofthe measure κ and the Cauchy-Schwarz inequality we obtain, for any realization of the field η ,FE τ,h Λ ( Rη ) + FE τ,h Λ ( ̃ Rη )= − β ∣ Λ ∣ ln ⎡⎢⎢⎢⎢⎣∫ S Λ0 exp (− βH Rη,h Λ ( σ )) ∏ v ∈ Λ κ ( dσ v ) ∫ S Λ0 exp (− βH ̃ Rη,h Λ ( σ )) ∏ v ∈ Λ κ ( dσ v )⎤⎥⎥⎥⎥⎦= − β ∣ Λ ∣ ln ⎡⎢⎢⎢⎢⎣∫ S Λ0 exp (− βH Rη,h Λ ( Rσ )) ∏ v ∈ Λ κ ( dσ v ) ∫ S Λ0 exp (− βH ̃ Rη,h Λ ( ̃ Rσ )) ∏ v ∈ Λ κ ( dσ v )⎤⎥⎥⎥⎥⎦≤ − β ∣ Λ ∣ ln ∫ S Λ0 exp ⎛⎜⎝− β H Rη,h Λ ( Rσ ) + H ̃ Rη,h Λ ( ̃ Rσ ) ⎞⎟⎠ ∏ v ∈ Λ κ ( dσ v ) . We then use the estimate (8.19) and obtainFE τ, ( Rη ) + FE τ, ( ̃ Rη ) ≤ − β ∣ Λ ∣ ln ∫ S Λ0 exp (− βH ,η Λ ( σ ) − βCl d − − βC(cid:96) d ∣ h ∣) ∏ v ∈ Λ κ ( dσ ) (8.20) ≤ FE τ, ( η ) + Cl d − L d + C(cid:96) d L d ∣ h ∣ . Moreover, by the definition of the rotations R and ̃ R , we have Rη = ̃ Rη in the box the box Λ, ( Rη ) i = ( ̃ Rη ) i = − η i along the i -th component of the field inside the box Λ, and Rη = ̃ Rη = η outside the annulus ( Λ ∖ ) . Using the rotational invariance of the law of the field η , weobtain the identities(8.21) E [ FE τ, ( R ⋅) ∣ η ( Λ ∖ )∪ Λ ,i ] = E [ FE τ, ( ̃ R ⋅) ∣ η ( Λ ∖ )∪ Λ ,i ] = E [ ̃ FE τ, , Λ ∣ η ( Λ ∖ )∪ Λ ,i ] . Taking the conditional expectation with respect to the field η ( Λ ∖ )∪ Λ ,i in the inequality (8.20)and using the identity (8.21) completes the proof of (8.9). (cid:3) Proof of Theorem 3.
In this section, we obtain an algebraic rate of convergence for theexpectation of the spatially and thermally averaged magnetization in the subcritical dimensions d = , ,
3. We prove the following more refined version of Theorem 3, which takes into accountthe dependence in the external magnetic field h . Theorem 3 is a direct consequence upon taking ∣ h ∣ ≤ L − . Theorem 5.
Let d ∈ { , , } , L ≥ be an integer, λ > and β > and a magnetic field h ∈ R n satisfying ∣ h ∣ ≤ . Let τ ∈ S ∂ Λ L be a boundary condition and set (cid:96) ∶= ∣ h ∣ − ∧ L . There exists aconstant C > depending on λ , n and Ψ such that, (8.22) RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ (cid:96) ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ C(cid:96) d − . Additionally, for any magnetic field h ∈ R n satisfying ∣ h ∣ ≤ , (8.23) E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L RRRRRRRRRRR ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ,h Λ L RRRRRRRRRRR⎤⎥⎥⎥⎥⎦ ≤ C (∣ h ∣ ∨ L − ) − d ( − d ) . Algebraic decay of the magnetization with fixed boundary condition.
In this section, wecombine Proposition 8.4 with Lemma 8.1 to obtain the algebraic decay of the magnetizationwith a fixed boundary condition stated in (8.22).
Proof of Theorem 5: estimate (8.22) . Let us fix an integer L ≥
2, a boundary condition τ ∈S ∂ Λ L , an external magnetic field h ∈ R n such that ∣ h ∣ ≤ i ∈ { , . . . , n } . Weintroduce the notation (cid:96) ∶= ∣ h ∣ − ∧ L . Applying Proposition 8.4 with Λ = Λ L and Λ = Λ (cid:96) , wehave the inequality(8.24) E [ ̃ FE τ,h Λ L , Λ (cid:96) − FE τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ] ≤ C (cid:96) d − L d P − almost-surely . Let us note that the conditional expectation depends only on the realization of the averagedfield ˆ η Λ ; it can thus be seen as a function defined on R and valued in R (see Section 6.3). Bythe definition of the free energy ̃ FE τ,h Λ L , Λ (cid:96) and the η → − η invariance of the law of the randomfield, we have the identity(8.25) E [ ̃ FE τ,h Λ L , Λ (cid:96) ∣ ˆ η Λ (cid:96) ,i ] ( ˆ η Λ (cid:96) ,i ) = E [ FE τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ] (− ˆ η Λ (cid:96) ,i ) . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 33
To ease the notation, let us define the map G ∶ R → R by the formula G ( ˆ η Λ (cid:96) ,i ) ∶= E [ FE τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ] (− ˆ η Λ (cid:96) ,i ) − E [ FE τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ] ( ˆ η Λ (cid:96) ,i ) . We note that, by Proposition 6.3 and the Gaussianity of the field, the derivative of the map G is explicit and we have(8.26) G ′ ( ˆ η Λ (cid:96) ,i ) = λ E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v,i ⟩ τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ⎤⎥⎥⎥⎥⎦ (− ˆ η Λ (cid:96) ,i ) + λ E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v,i ⟩ τ,h Λ L ∣ ˆ η Λ (cid:96) ,i ⎤⎥⎥⎥⎥⎦ ( ˆ η Λ (cid:96) ,i ) . The strategy is to apply Lemma 8.1 with the map g ∶ R → R defined by the formula(8.27) g ( ˆ η Λ (cid:96) ,i ) ∶= L d C(cid:96) d − ∣ Λ (cid:96) ∣ / G ′ ⎛⎝ ˆ η Λ (cid:96) ,i ∣ Λ (cid:96) ∣ / ⎞⎠ , where C is the constant appearing in the right side of (8.24). Let us first verify that the map g belongs to the set G introduced in (8.1). We fix an interval I = [ t , t ] ⊆ R . By the identity (8.26)and the inequality (8.24), we have ∣∫ I g ( t ) dt ∣ = L d C(cid:96) d − RRRRRRRRRRR G ⎛⎝ t ∣ Λ (cid:96) ∣ / ⎞⎠ − G ⎛⎝ t ∣ Λ (cid:96) ∣ / ⎞⎠RRRRRRRRRRR ≤ . Consequently, the map g belongs to the set G . We can thus apply Lemma 8.1 and obtain(8.28) ∣∫ R g ( t ) e − t dt ∣ ≤ C, for some numerical constant C >
0. Using the definition of g stated in (8.27) and performingthe change of variable t → ∣ Λ (cid:96) ∣ t , we obtain the inequality ∣∫ R G ′ ( t ) e − ∣ Λ (cid:96) ∣ t dt ∣ ≤ C (cid:96) d − L d , for some numerical constant C . Using that the random variable ˆ η Λ (cid:96) ,i is Gaussian of variance ∣ Λ (cid:96) ∣ − and the identity (8.26), we obtain the equality √ ∣ Λ (cid:96) ∣ π ∫ R G ′ ( t ) e − ∣ Λ (cid:96) ∣ t dt = λ E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v,i ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦ . Combining the two previous displays shows(8.29)
RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ (cid:96) ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v,i ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ C(cid:96) d − . Since the inequality (8.29) holds for any i ∈ { , . . . , n } , it implies the inequality (8.22). (cid:3) Algebraic decay of the magnetization uniform over the boundary conditions.
In thissection, we use the results established in Section 8.3.1 to obtain an algebraic decay for themagnetization which is uniform over the boundary condition.
Proof of Theorem 5: estimate (8.23) . Fix a side length L ≥
2. We consider the system withperiodic boundary condition and note that, for any vertex v ∈ Λ L and any h ∈ R n ,(8.30) E [⟨ σ v ⟩ per ,h Λ L ] = E [⟨ σ ⟩ per ,h Λ L ] . Let us now fix h ∈ R n such that ∣ h ∣ ≤ (cid:96) ∶= ∣ h ∣ − ∧ L . Applying Theorem 5 with theboxes Λ L and Λ (cid:96) , and using (8.30), we obtain ∣ E [⟨ σ ⟩ per ,h Λ L ]∣ = RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ (cid:96) ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v ⟩ per ,h Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ C(cid:96) d − ≤ C (∣ h ∣ ∨ L − ) − d Using the identity (6.6), we obtain, for any h ∈ R n such that ∣ h ∣ ≤ ∣ E [ FE per ,h Λ L − FE per , L ]∣ ≤ n ∑ i = ∫ RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ per ,th Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ∣ h i ∣ dt (8.31) ≤ n ∑ i = ∫ ∣ E [⟨ σ ,i ⟩ per ,th Λ L ]∣ ∣ h i ∣ dt ≤ C ∣ h ∣ (∣ h ∣ ∨ L − ) − d ≤ C (∣ h ∣ ∨ L − ) − d , where we used ∣ h ∣ ≤ ∣ h ∣ ∨ L − in the last inequality. Let us then fix an integer i ∈ { , . . . , n } . Foreach realization of the random field η and each h ∈ R n , we let τ i ( η, h ) ∈ S ∂ Λ L be a boundarycondition satisfying(8.32) 1 ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i ( η,h ) ,h Λ L = sup τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L . Note that, by the inequality (6.8) of Proposition 6.3 (with R = h ∈ R n ,(8.33) ∣ E [ FE τ i ( η,h ) ,h Λ L ] − E [ FE per ,h Λ L ]∣ ≤ CL and ∣ E [ FE τ i ( η,h ) , L ] − E [ FE per , L ]∣ ≤ CL .
A combination of the inequalities (8.31) and (8.33) yields, for any h ∈ R n satisfying ∣ h ∣ ≤ ∣ E [ FE τ ( η,h ) ,h Λ L ] − E [ FE τ ( η,h ) , L ]∣ ≤ C (∣ h ∣ ∨ L − ) − d + CL .
Let us fix h = ( h , . . . , h n ) ∈ R n such that ∣ h ∣ ≤
1, set α ∶= /( − d ) and denote by ̃ h ∶= ( h , . . . , h i − , h i + (∣ h ∣ ∨ L − ) α , h i + , . . . , h n ) . We note that we have ∣̃ h ∣ ≤ (∣ h ∣ ∨ L − ) α ≤
2. We next introduce the function G ∶ h ′ ↦ − E [ FE τ ( η,h ) ,h ′ Λ L ] . Observe that the map G is convexe and that its derivative with respect to the i − th variablesatisfies ∂G∂h ′ i ( h ′ ) = E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ ( η,h ) ,h ′ Λ L ⎤⎥⎥⎥⎥⎦ , and that, by (8.34) and the inequalities of ∣̃ h ∣ ≤ (∣ h ∣ ∨ L − ) α and L ≥ (∣ h ∣ ∨ L − ) − / , ∣ G (̃ h ) − G ( h )∣ ≤ C (∣̃ h ∣ ∨ L − ) − d + C (∣ h ∣ ∨ L − ) − d + CL ≤ C (∣ h ∣ ∨ L − ) α ( − d ) + C (∣ h ∣ ∨ L − ) . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 35
Combining the previous observations with (8.32) and (8.34) and using the value α = /( − d ) ,we obtain E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦ = E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i ( η,h ) ,h Λ L ⎤⎥⎥⎥⎥⎦ = ∂G∂h ′ i ( h )≤ G (̃ h ) − G ( h )(∣ h ∣ ∨ L − ) α (8.35) ≤ C (∣ h ∣ ∨ L − ) − d ( − d ) . To complete the argument, let us consider the random boundary condition τ i, − ( η, h ) defined soas to satisfy(8.36) 1 ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i, − ( η,h ) ,h Λ L = inf τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L , and define ̃ τ i, − ( η, h ) ∶= − τ i (− η, h ) . Using a similar computation as the one performed in (8.35),we obtain(8.37) E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ ̃ τ i, − ( η,h ) , − h Λ L ⎤⎥⎥⎥⎥⎦ ≤ C (∣ h ∣ ∨ L − ) − d ( − d ) , Combining (8.36), (8.37) with the identity (6.11) yields(8.38) E ⎡⎢⎢⎢⎢⎣ inf τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦ = − E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ ̃ τ i ( η,h ) , − h Λ L ⎤⎥⎥⎥⎥⎦ ≥ − C (∣ h ∣ ∨ L − ) − d ( − d ) . Combining (8.35) and (8.38) implies(8.39) E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L ∣ Λ L ∣ RRRRRRRRRRR ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L RRRRRRRRRRR⎤⎥⎥⎥⎥⎦ ≤ C (∣ h ∣ ∨ L − ) − d ( − d ) . Using that the inequality (8.39) holds for any integer i ∈ { , . . . , n } completes the proof of theestimate (8.23). (cid:3) Proof of Theorem 4.
The objective of this section is to prove Theorem 4 following theoutline presented at the beginning of Section 8.8.4.1.
A lower bound on the conditional expectation of the spatially averaged magnetization.
Inthe first step of the proof, we show that that, if the averaged value of the field η in a box Λ isnegative enough, then the thermally and spatially averaged magnetization of the continuousspin system with periodic boundary conditions in the box Λ must be small. The argument relieson a combination of the variational lemma stated in Lemma 8.2 and of the Mermin-Wagnerupper bound for the free energy (Proposition 8.4). Before stating the result, we introduce anotation for the quantile of the normal distribution which will be used in the statement andproof of Lemma 8.6. Definition 8.5 (Quantile of the normal distribution) . For each δ >
0, we define the exp (− δ ) -quantile of the normal distribution by the formula(8.40) t δ ∶= min { t ∈ R ∶ √ π ∫ ∞ t e − s ds ≤ − exp (− δ )} , Let us note that there exist two constants c, C ∈ ( , ∞) such that for any δ ∈ ( , / ] , − Cδ − ≤ t δ ≤ − cδ − . Lemma 8.6.
Let d = . Fix β > , λ > and i ∈ { , . . . , n } and a box Λ ⊆ Z d . Let τ ∈ S ∂ Λ bea boundary condition (which may be the free or periodic boundary conditions). For any box Λ ofside length (cid:96) such that ⊆ Λ , any δ > , h ∈ R n satisfying ∣ h ∣ ≤ (cid:96) , we have the estimate E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ≤ Cδ on the event { ˆ η Λ ,i ≤ (cid:96) − t δ } . Proof.
We denote by L the side length of the box Λ . By Proposition 8.4 and using theassumption ∣ h ∣ ≤ / (cid:96) , we have the inequality(8.41) E [ ̃ FE τ,h Λ , Λ − FE τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ≤ C(cid:96) L P − almost surely . As in (8.25), we have the identity E [ ̃ FE τ,h Λ , Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) = E [ FE τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] (− ˆ η Λ ,i , η ( Λ ∖ ) ,i ) . We first claim that there exists a constant C > δ > P [ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ≤ Cδ ∣ η ( Λ ∖ ) ,i ] ≥ exp (− δ ) P − almost surely . To prove the inequality (8.42), we introduce the following map G Λ ∶ ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) ↦ E [ FE τ,h Λ , Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] (− ˆ η Λ ,i , η ( Λ ∖ ) ,i )− E [ FE τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) . Let us note that the map G Λ satisfies the identity ∂G Λ ∂ ˆ η Λ ,i ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) = λ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) (8.43) + λ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] (− ˆ η Λ ,i , η ( Λ ∖ ) ,i ) . We next show the following inequality: for every δ > P ( ∂G Λ ∂ ˆ η Λ ,i < C(cid:96) L δ ∣ η ( Λ ∖ ) ,i ) ≥ exp (− δ ) P − almost-surely . To prove the estimate (8.44), we fix a realization of the field η ( Λ ∖ ) ,i , and apply Proposition 8.2with the choice of function g ( ˆ η Λ ,i ) ∶= L C ( ∨ λ ) (cid:96) ∂G Λ ∂ ˆ η Λ ,i ( ˆ η Λ ,i (cid:96) , η ( Λ ∖ ) ,i ) , where the constant C is the one which appears in the right side of (8.41). We first verify thatthe map g belongs to the set G (defined in (8.1)). The result is a consequence of the followingcomputation: by (8.41), we have, for any interval I = [ t , t ] ⊆ R , ∣∫ I g ( t ) dt ∣ = L C ( ∨ λ ) (cid:96) ∣ G Λ ( t (cid:96) , η ( Λ L ∖ ) ,i ) − G Λ ( t (cid:96) , η ( Λ ∖ ) ,i )∣ ≤ . The fact that the map g is larger than − S n − (and thus the norm of a spin is always equal to 1).Applying Proposition 8.2 yields, for any δ > ∫ R { g ( t )≤ δ } e − t / dt ≥ e − Cδ . UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 37
Rescaling the previous inequality, using that the averaged field ˆ η Λ ,i is Gaussian of variance (cid:96) − ,and that it is independent of the field η ( Λ ∖ ) ,i gives the estimate (8.44). We then reformulatethe inequality (8.44): using the formula (8.43) and a union bound, we obtain that there existsa constant C > P − almost surely,(8.45) P [ E [ (cid:96) ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ≤ Cδ ∣ η ( Λ ∖ ) ,i ]+ P [ E [ (cid:96) ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ L ∖ ) ,i ] (− ˆ η Λ ,i , η ( Λ ∖ ) ,i ) ≤ Cδ ∣ η ( Λ ∖ ) ,i ] ≥ exp (− δ ) . Since the law of the random variable ˆ η Λ ,i is invariant under the involution ˆ η Λ ,i → − ˆ η Λ ,i , the twoterms in the left side of (8.45) are equal. We thus obtain, for any δ > P [ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ≤ Cδ ∣ η ( Λ ∖ ) ,i ] ≥
12 exp (− δ ) P − a.s.Since the estimate (8.46) is valid for any δ >
0, it implies the inequality (8.42) by increasing thevalue of the constant C if necessary. We then observe that, for each fixed realization of the field η ( Λ ∖ ) ,i , the map ˆ η Λ ,i ↦ − E [ FE τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) is convex and that its derivative is the functionˆ η Λ ,i ↦ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) . Since the derivative of a convex function is increasing, we obtain that the mapˆ η Λ ,i ↦ E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) is increasing. Combining this observation with the inequality (8.46), the definition of thequantile t δ stated in (8.40), and the fact that the random variable ˆ η Λ ,i is Gaussian of variance (cid:96) − , we obtain, for any δ > E [ ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ,i , η ( Λ ∖ ) ,i ] ( ˆ η Λ ,i , η ( Λ ∖ ) ,i ) ≤ Cδ on the event { ˆ η Λ ,i ≤ (cid:96) − t δ } . The proof of Lemma 8.6 is complete. (cid:3)
Mandelbrot percolation argument.
In this section, we combine the result of Lemma 8.6with a Mandelbrot percolation argument to obtain a quantitative rate of convergence on theexpected value (in the random field) of the spatially and thermally averaged magnetization witha fixed boundary condition.
Lemma 8.7.
Let d = . Fix β > , λ > , a box Λ ⊆ Z d , an integer L ≥ such that Λ L ⊆ Λ ,and an external magnetic field h ∈ R n satisfying ∣ h ∣ ≤ L − . Let τ ∈ S ∂ Λ be a boundary condition(which may be the free and periodic boundary conditions). There exists a constant C > depending only on λ , m and Ψ such that (8.48) RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ,h Λ ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ C √ ln ln L .
Proof of Theorem 4.
First, let us note that, by the identity (6.11), it is sufficient, in order toprove (8.48), to prove, for any integer i ∈ { , . . . , n } ,(8.49) E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ ⎤⎥⎥⎥⎥⎦ ≤ C √ ln ln L .
Additionally, it is sufficient to prove the inequality (8.49) when L is large enough.We now fix an integer i ∈ { , . . . , n } and prove the inequality (8.49). To this end, we set δ = C /( ln ln L ) ∧ C whose value is decided at the end of the proof.The strategy is to implement a Mandelbrot percolation argument in the box Λ L with thefollowing definition of good box:a box Λ ⊆ Λ L of side length (cid:96) is good if ˆ η Λ ,i ≤ (cid:96) − t δ .We let k ∶= ⌊ exp ( √ ln L )⌋ , assume that L is large enough so that k ≥
4, and denote by l max thelargest integer which satisfies k l max ≤ √ L . We introduce the set of boxes T l ∶= {( z + [− Lk l , Lk l ) d ) ∩ Z d ∶ z ∈ Lk l Z d ∩ Λ L } , and implement the Mandelbrot percolation argument developed in the second step of the proof ofLemma 7.7. We obtain a collection Q ⊆ ∪ l max l = T l of good boxes. We need to prove two propertiespertaining to the collection Q . First we show that the set of uncovered points is typically small.To this end, we prove the inequality, for any vertex v ∈ Λ L ,(8.50) P ( v is not covered ) ≤ exp (− c √ ln L ) . Second, we need to prove that the expected magnetization on a box of the collection Q is small;specifically, we show the estimate, for each box Λ ⊆ Λ L ,(8.51) E [ { Λ ∈Q} ∣ Λ ∣ ∑ v ∈ Λ ⟨ σ v,i ⟩ τ,h Λ ] ≤ δ E [ { Λ ∈Q} ] + C ( ln L ) . We first focus on the proof of the inequality (8.50). To this end, we fix a vertex v ∈ Λ L , letΛ ( v ) , . . . , Λ l max ( v ) be the boxes of the collections T , . . . , T l max containing the vertex v , anddenote their side length by (cid:96) , . . . , (cid:96) max respectively. For any l ∈ { , . . . , l max − } , we denote by k l ∶= (cid:96) l / (cid:96) l + the ratio between the two side length (cid:96) l and (cid:96) l + and note that there exist numericalconstants c, C such that ck ≤ k l ≤ Ck as soon as L is large enough. We denote byˆ η l ∶= ∣ Λ l ( v )∣ ∑ u ∈ Λ l ( v )∖ Λ l + ( v ) η u,i . Note that the random variables ˆ η Λ l ( v ) ,i and ˆ η l are typically close from each other: the law ofthe random variable ˆ η l − ˆ η Λ l ( v ) ,i is Gaussian and its variance is equal to 1 /( k l (cid:96) l ) . We also notethat the random variable ˆ η l is independent of the restriction field η to the box Λ l + ( v ) .We have the identity of events(8.52) { v is not covered } = l max ⋂ l = { ˆ η Λ l ( v ) ,i > (cid:96) − l t δ } . We then show that the l max events on the right side of (8.52) are well-approximated byindependent events, and use the independence to estimate the probability of their intersection.To this end, we use the identity ˆ η l + ˆ η Λ l + ( v ) ,i k l = ˆ η Λ l ( v ) ,i , and note that the following inclusion UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 39 holds(8.53) l max ⋂ l = { ˆ η Λ l ( v ) ,i ≥ (cid:96) − l t δ }⊆ ( l max − ⋂ l = { ˆ η l ≥ (cid:96) − l ( t δ − δ )} ⋂ { ˆ η Λ l max ( v ) ,i ≥ (cid:96) − l max t δ }) ⋃ ( l max ⋃ l = { ˆ η Λ l ( v ) ,i ≥ k l δ(cid:96) − l }) . Using that the random variables ( ˆ η l ) ≤ l ≤ l max − are independent and a union bound, we obtain(8.54) P ( l max ⋂ l = { ˆ η Λ l ( v ) ,i ≥ (cid:96) − j t δ }) ≤ l max − ∏ l = P ( ˆ η l ≥ (cid:96) − l ( t δ − δ ))·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.54) −( i ) + l max ∑ l = P ( ˆ η Λ l ( v ) ,i ≥ k l δ(cid:96) l )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.54) −( ii ) . We estimate the terms (8.54)-(i) and (8.54)-(ii) separately. For the term (8.54)-(i), we notethat, if L is chosen large enough, the quantile t δ satisfies the inequality − c / δ ≤ t δ ≤ − C / δ . Anexplicit computation shows that there exist two constant c, C ∈ ( , ∞) such that P ( ˆ η l ≥ (cid:96) − l ( t δ − δ )) ≤ − c exp (− Cδ ( + δ ) ) ≤ − c exp (− Cδ ) . We recall that we have set k = ⌊ exp ( √ ln L )⌋ , δ = C /( ln ln L ) ∧
2, and l max ∶= ⌊ L / ln k ⌋ ≃( ln L ) / . Consequently, if the constant C is chosen large enough,(8.55) l max − ∏ l = P ( ˆ η l ≥ (cid:96) − l ( t δ − δ )) ≤ ( − c exp (− Cδ )) l max − ≤ exp (− c √ ln L ) . We now estimate the term (8.54)-(ii). The lower bound k l ≥ ck and an explicit computationgives, for each l ∈ { , . . . , l max } , P ( ˆ η Λ l ,i ≥ k l (cid:96) − l δ ) ≤ exp (− ck δ ) , and thus(8.56) l max ∑ l = P ( ˆ η Λ l ,i ≥ k(cid:96) − l δ ) ≤ l max exp (− ck δ ) ≤ exp (− c √ ln L ) . A combination of (8.54), (8.55) and (8.56) implies (8.52).We now focus on the proof of the inequality (8.51). We fix an integer l ∈ { , . . . , l max } , considera box Λ ′ ∈ T l and denote its side length by (cid:96) l . We denote by Λ ′ , . . . , Λ ′ l − the family of boxeswhich contain the box Λ ′ and belong to the sets T , . . . , T l − respectively. We denote the sidelength of these boxes by (cid:96) , . . . , (cid:96) l − . By construction of the collection Q , we have the identity { Λ ′ ∈ Q} = { ˆ η Λ ′ ,i ≤ (cid:96) − l t δ } ⋂ l − ⋂ j = { ˆ η Λ ′ j ,i > (cid:96) − j t δ } . Our objective is to prove that this event is well-approximated by an event which belongs to thesigma-algebra generated by the random variables ˆ η Λ ′ ,i and η Λ ∖ ′ ,i . To this end, let us define,for any integer j ∈ { , . . . , l − } , ˆ η ′ j ∶= ∣ Λ ′ j ∣ ∑ u ∈ Λ ′ j ∖ ′ η u,i . Let us note that the random variable ˆ η ′ j depends on the realization of the field outside the box2Λ ′ , and that it satisfies the identity ˆ η Λ ′ j = ˆ η ′ j + ∣ ′ ∣∣ Λ ′ j ∣ ˆ η ′ ,i . We additionally note that, by the definitions of the cube Λ ′ j , the ratio between the volumes ofthe boxes Λ ′ j and 2Λ ′ is at least of order k : there exists a constant c such that ∣ Λ ′ j ∣ ≥ ck ∣ Λ ′ ∣ . As a consequence of the previous definitions and observations, we have the inclusion(8.57) { ˆ η Λ ′ j ,i > (cid:96) − j t δ } ∆ { ˆ η ′ j > (cid:96) − j t δ } ⊆ ⎧⎪⎪⎨⎪⎪⎩∣ ˆ η ′ j − (cid:96) − j t δ ∣ ≤ (cid:96) j ( ln L ) ⎫⎪⎪⎬⎪⎪⎭·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.57) −( i ) ∪ ⎧⎪⎪⎨⎪⎪⎩∣ ˆ η ′ ,i ∣ ≥ ck (cid:96) l ( ln L ) ⎫⎪⎪⎬⎪⎪⎭·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.57) −( ii ) , where the symbol ∆ denotes the symmetric difference between the events { ˆ η Λ ′ j ,i > (cid:96) − j t δ } and { ˆ η ′ j > (cid:96) − j t δ } . We estimate the probabilities of the two events in the right side of (8.57). For theevent (8.57)-(i), we note that the random variable ˆ η ′ j is Gaussian and that its variance satisfiesvar ˆ η ′ j = ∣ Λ ′ j ∖ ′ ∣∣ Λ ′ j ∣ ≥ ∣ Λ ′ j ∣ = (cid:96) j , where we have used the inequality ∣ Λ ′ j ∖ ′ ∣ ≥ ∣ Λ ′ j ∣ /
2, which is a consequence of the definitionof the box Λ ′ j together with the assumption k ≥
4. We obtain(8.58) P ⎛⎝RRRRRRRRRRR ˆ η ′ j − t δ (cid:96) j RRRRRRRRRRR ≤ (cid:96) j ( ln L ) ⎞⎠ ≤ P ⎛⎝∣ ˆ η ′ j ∣ ≤ (cid:96) j ( ln L ) ⎞⎠ ≤ C ( ln L ) . For the event (8.57)-(ii), we use that the random variable ˆ η ′ is Gaussian and that its varianceis equal to (cid:96) − l to write(8.59) P ⎛⎝∣ ˆ η ′ ,i ∣ ≥ ck (cid:96) l ( ln L ) ⎞⎠ ≤ C exp (− ck ( ln L ) ) ≤ C ( ln L ) . This result implies(8.60) { Λ ′ ∈ Q} ∆ ⎛⎝{ ˆ η Λ ′ ,i ≤ (cid:96) − l t δ } ⋂ l ⋂ j = { ˆ η ′ j > (cid:96) − j t δ }⎞⎠⊆ ⎧⎪⎪⎨⎪⎪⎩∣ ˆ η ′ ,i ∣ ≥ ck (cid:96) l ( ln L ) ⎫⎪⎪⎬⎪⎪⎭ ⋃ l − ⋃ j = ⎧⎪⎪⎨⎪⎪⎩∣ ˆ η ′ j − (cid:96) − j t δ ∣ ≤ (cid:96) j ( ln L ) ⎫⎪⎪⎬⎪⎪⎭ . Let us introduce the notation E Λ ′ ∶= { ˆ η Λ ′ ,i ≤ (cid:96) − l t δ } ⋂ l − ⋂ j = { ˆ η ′ j > (cid:96) − j t δ } . A consequence of the inclusion (8.60) is the inequality of indicator functions(8.61) ∣ { Λ ′ ∈Q} − E Λ ′ ∣ ≤ {∣ ˆ η Λ ′ ,i ∣≥ ck (cid:96) l ( ln L ) } + l − ∑ j = {∣ ˆ η ′ j − (cid:96) − j t δ ∣≤ (cid:96) j ( ln L ) } . We note that the event E Λ ′ is measurable with respect to the σ -algebra generated by the randomvariables ˆ η Λ ′ ,i and η Λ ∖ ′ ,i . Using Lemma 8.6 and the fact that the event E Λ ′ is contained in UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 41 the event { ˆ η Λ ′ ,i ≤ (cid:96) − l t δ } , we see that E [ E Λ ′ ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ] = E [ E [ E Λ ′ ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ′ ,i , η Λ L ∖ ′ ,i ]] (8.62) = E [ E Λ ′ E [ ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ∣ ˆ η Λ ′ ,i , η Λ L ∖ ′ ,i ]]≤ δ E [ E Λ ′ ]= δ E [ E Λ ′ ] . We can now conclude the proof of the inequality (8.51). Applying the estimates (8.58), (8.59), (8.61),the computation (8.62), and the upper bound l ≤ ln L , we obtain E [ { Λ ′ ∈Q} ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ]≤ E [ E Λ ′ ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ] + E ⎡⎢⎢⎢⎢⎢⎣ {∣ ˆ η ′ ,i ∣≥ ck (cid:96) l ( ln L ) } + l − ∑ j = {∣ ˆ η ′ j − tδ(cid:96) j ∣≤ (cid:96) j ( ln L ) } ⎤⎥⎥⎥⎥⎥⎦≤ δ E [ E Λ ′ ] + E ⎡⎢⎢⎢⎢⎢⎣ {∣ ˆ η ′ ,i ∣≥ ck (cid:96) l ( ln L ) } + l − ∑ j = {∣ ˆ η ′ j − tδ(cid:96) j ∣≤ (cid:96) j ( ln L ) } ⎤⎥⎥⎥⎥⎥⎦≤ δ E [ { Λ ′ ∈Q} ] + E ⎡⎢⎢⎢⎢⎢⎣ {∣ ˆ η ′ ,i ∣≥ ck (cid:96) l ( ln L ) } + l − ∑ j = {∣ ˆ η ′ j − tδ(cid:96) j ∣≤ (cid:96) j ( ln L ) } ⎤⎥⎥⎥⎥⎥⎦≤ δ E [ { Λ ′ ∈Q} ] + Cl ( ln L ) ≤ δ E [ { Λ ′ ∈Q} ] + C ( ln L ) . The proof of (8.51) is complete.We now use the two properties (8.50) and (8.51) of the collection Q of good boxes to completethe proof of Theorem 4. We write ∑ Λ ′ ⊆ Λ L to refer to the sum ∑ l max l = ∑ Λ ′ ∈T l . We decompose theexpectation and write E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ ⎤⎥⎥⎥⎥⎦ (8.63) = E ⎡⎢⎢⎢⎢⎣ ∑ Λ ′ ⊆ Λ L ∣ Λ ′ ∣∣ Λ L ∣ { Λ ′ ∈Q} ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ⎤⎥⎥⎥⎥⎦ + E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L { v is uncovered } ⟨ σ v,i ⟩ τ,h Λ ⎤⎥⎥⎥⎥⎦= ∑ Λ ′ ⊆ Λ L ∣ Λ ′ ∣∣ Λ L ∣ E [ { Λ ′ ∈Q} ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ]·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.63) −( i ) + ∣ Λ L ∣ ∑ v ∈ Λ L E [ { v is uncovered } ⟨ σ v,i ⟩ τ,h Λ ] . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (8.63) −( ii ) We then estimate the two terms in the right side separately. We begin with the term (8.63)-(i),use the inequality (8.51) and the observations ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣ { Λ ∈Q} ≤ ∣ Λ ∣ and ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣ = l max ∑ l = ∑ Λ ′ ∈T l ∣ Λ ′ ∣ = ( l max + ) ∣ Λ ∣ ≤ C ln L ∣ Λ ∣ . We obtain ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣∣ Λ ∣ E [ { Λ ′ ∈Q} ∣ Λ ′ ∣ ∑ v ∈ Λ ′ ⟨ σ v,i ⟩ τ,h Λ ] ≤ δ ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣∣ Λ ∣ E [ { Λ ′ ∈Q} ] + C ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣∣ Λ ∣ ( ln L ) ≤ δ ∣ Λ ∣ E [ ∑ Λ ′ ⊆ Λ ∣ Λ ′ ∣ { Λ ′ ∈Q} ] + C ln L ( ln L ) ≤ Cδ.
There only remains to treat the term (8.63)-(ii). We use to the estimate (8.50) and write ∣ ∣ Λ ∣ ∑ v ∈ Λ E [ { v is uncovered } ⟨ σ v,i ⟩ τ,h Λ ]∣ ≤ ∣ Λ ∣ ∑ v ∈ Λ P [ v is uncovered ]≤ exp (−√ ln L )≤ C √ ln ln L .
A combination of the two previous displays with the identity (8.63) completes the proof ofLemma 8.7. (cid:3)
Proof of Theorem 4.
In this section, we combine the result of Lemma 8.7 (applied withperiodic boundary condition) with an argument similar to the one developed in Section 8.3.2 tocomplete the proof of Theorem 4.
Proof of Theorem 4.
Fix a side length L ≥ h ∈ R n such that ∣ h ∣ ≤ /
4, and set (cid:96) ∶= L / ∧ ∣ h ∣ − / .We consider the system with periodic boundary condition and apply Lemma 8.7 with the boxesΛ ∶= Λ L and Λ L = Λ (cid:96) . We obtain ∣ E [⟨ σ ⟩ per ,h Λ L ]∣ = RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ (cid:96) ∣ ∑ v ∈ Λ (cid:96) ⟨ σ v ⟩ per ,h Λ L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ C √ ln ln (cid:96) ≤ C √ ln ln ( L ∧ ∣ h ∣ − ) , where we used the inequality ln (cid:96) ≥ c ln ( L ∧ ∣ h ∣ − ) . Integrating over h as it was done in (8.31),we deduce that(8.64) ∣ E [ FE per ,h Λ L − FE per , L ]∣ ≤ C ∣ h ∣√ ln ln ( L ∧ ∣ h ∣ − ) . Let us then fix an integer i ∈ { , . . . , n } . For each realization of the random field η and each h ∈ R n , we let τ i ( η, h ) ∈ S ∂ Λ L be a boundary condition satisfying(8.65) 1 ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i ( η,h ) ,h Λ L = sup τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i ,h Λ L . Using the inequality (6.8) of Proposition 6.3, we have, for any h ∈ R n ,(8.66) ∣ E [ FE τ i ( η,h ) ,h Λ L ] − E [ FE per ,h Λ L ]∣ ≤ CL and ∣ E [ FE τ i ( η,h ) , L ] − E [ FE per , L ]∣ ≤ CL .
A combination of the inequalities (8.64) and (8.66) yields, for any h ∈ R m satisfying ∣ h ∣ ≤ / ∣ E [ FE τ ( η,h ) ,h Λ L ] − E [ FE τ ( η,h ) , L ]∣ ≤ C ∣ h ∣√ ln ln ( L ∧ ∣ h ∣ − ) + CL .
We then fix ∣ h ∣ ≤
1, and denote by ̃ h ∶= ( h , . . . , h i − , h i + √∣ h ∣ ∨ L − , h i + , . . . , h m ) . We note that ∣̃ h ∣ ≤ √∣ h ∣ ∨ L − ≤ UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 43
We introduce the function G ∶ h ′ ↦ − E [ FE τ ( η,h ) ,h ′ Λ L ] , observe that the map G is convex, andthat its derivative with respect to the i − th variable satisfies ∂G∂h ′ i ( h ′ ) = E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ ( η,h ) ,h ′ Λ L ⎤⎥⎥⎥⎥⎦ . Additionally, by (8.67) and the definition of ̃ h , we have ∣ G (̃ h ) − G ( h )∣ ≤ C √ L − ∨ ∣ h ∣√ ln ln L ∧ ∣ h ∣ − + CL ≤ C √ L − ∨ ∣ h ∣√ ln ln ( L ∧ ∣ h ∣ − ) . Combining the previous observations with (8.65) and (8.67), we obtain E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ i ( η,h ) ,h Λ L ⎤⎥⎥⎥⎥⎦ ≤ G (̃ h ) − G ( h )√∣ h ∣ ∨ L − ≤ C √ ln ln ( L ∧ ∣ h ∣ − ) . Using the definition of the random boundary condition τ i ( η, h ) , we obtain E ⎡⎢⎢⎢⎢⎣ sup τ ∈S ∂ Λ L ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v,i ⟩ τ,h Λ L ⎤⎥⎥⎥⎥⎦ ≤ C √ ln ln ( L ∧ ∣ h ∣ − ) . The proof of Theorem 4 can be completed by using the same arguments as the one presented inthe proof of the estimate (8.23) of Theorem 3 in Section 8.3.2, we thus omit the details. (cid:3) Discussion and open problems
This work initiates the study of quantitative versions of the Aizenman–Wehr [5, 6] resulton the Imry–Ma rounding phenomenon [40]. In this section we discuss some of the remainingproblems:
Uniqueness conjecture:
As discussed in Section 2, we believe that a stronger qualitativestatement than the one provided by Aizenman–Wehr [5, 6] is valid. Namely, that in twodimensions the thermal averages ⟨ f v ( σ )⟩ τ Λ L cannot be significantly altered by changing τ inthe sense of Conjecture 2.1. We again point out that Conjecture 2.1 would imply as a specialcase the well-known belief that the two-dimensional Edwards-Anderson spin glass model hasa unique ground-state pair. It would be very interesting to make additional progress in thisdirection.It is also possible that analogous uniqueness statements hold in dimensions d = , d = Quantitative decay rate and possible phase transitions:
It is very natural to seek theoptimal rates in our quantitative results. We first discuss the general two-dimensional disorderedspin systems of Section 2. When the base Hamiltonian has finite range, exponential decayof correlations and uniqueness of the infinite-volume Gibbs measure follow in all dimensionsin the high-temperature regime ( β ≪ λ ≫ ) , from the disagreement percolation methods of van den Berg–Maes [54] andtheir adaptations by Gielis–Maes [37] (suitable versions of Dobrushin’s condition [30] shouldalso be applicable). The main interest is thus in the low-temperature (or even zero temperature)regime. We then believe that the correct rate in two dimensions should be much faster than theinverse power of log-log rate obtained in Theorem 1. Without assuming translation invariance,the decay cannot hold at a faster than power-law rate (e.g., as the noised observables ( f v ) canbe identically zero at all but one vertex), but it is possible that such a rate indeed holds in general. In the translation-invariant setup, it is even possible that exponential decay holds(noting that for a faster than power-law decay one needs to perform the spatial average in (2.7)over a smaller domain, say Λ L / , to avoid boundary effects), as proved for the nearest-neighborferromagnetic random-field Ising model [29, 28, 3] (though exponential decay is still open forthe non-nearest-neighbor version for which only a power-law upper bound is known [4]).We proceed to discuss the spin systems with continuous symmetry of Section 3. Exponentialdecay in the high-temperature regime ( β ≪
1) again follows in all dimensions [54, 37]. Thisis also expected in the strong disorder regime ( λ ≫ d = ,
3, arguments have beengiven both for [1, 53] and against [32, 38] this possibility in the physics literature. Power-lawdecay would imply a transition of the Berezinskii–Kosterlitz–Thouless type [8, 9, 41, 42, 34] asthe temperature or disorder strength varies and would thus be of great interest.
Higher-order continuous symmetries:
The form of continuous-symmetry that the spin O ( n ) model enjoys is that its formal Hamiltonian H ( σ ) ∶= ∑ v ∼ w ∣ σ v − σ w ∣ , where σ ∶ Z d → S n − , satisfies H ( Rσ ) = H ( σ ) for the operation R which rotates all spins in σ by the same angle. One can also envision spin systems enjoying higher-order symmetries in thesense that we now explain for the n = n > S in C and write σ v = e iθ v (where theangle θ is defined modulo 2 π ). For a polynomial P ∶ R d → R and configuration σ ∶ Z d → S definethe ‘polynomial rotation’ R P ( σ ) by R P ( σ ) v = e i ( θ v + P ( v )) . A Hamiltonian H on configurations σ ∶ Z d → S is then said to enjoy a continuous symmetry of order k if H ( R P ( σ )) = H ( σ ) for allpolynomials P ∶ R d → R of degree at most k . In particular, the case k = H ( σ ) ∶= ∑ v cos (( ∆ θ ) v ) where ∆ is the discrete Laplacian operator: ( ∆ θ ) v ∶= ∑ w ∶ w ∼ v ( θ w − θ v ) . Similarly, an exampleof a Hamiltonian enjoying continuous symmetry of order k = (cid:96) is obtained by replacing ∆with ∆ (cid:96) (the composition of ∆ with itself (cid:96) times) in (9.1) and an example enjoying continuoussymmetries of order k = (cid:96) + H ( σ ) ∶= ∑ v ∼ w cos (( ∆ (cid:96) θ ) v − ( ∆ (cid:96) θ ) w ) We do not know if these spin models have received attention in the literature.Higher-order symmetries reduce the surface tension of finite-range spin systems with smoothenergy. Specifically, we believe that an analog of Proposition 8.4 holds for a spin system havinga smooth, finite-range Hamiltonian enjoying a higher-order symmetry of order k with the factor (cid:96) d − replaced by (cid:96) d − k − . To prove this fact, one may follow the step of Proposition 8.4 with thefollowing modifications: ● References to Ψ should be replaced by corresponding references to the Hamiltonian. ● The spin wave in equation (8.10) needs to be replaced by a function θ satisfying θ ≡ Z d ∖ θ ≡ π in 2Λ and all discrete derivatives of order k + ψ are uniformly bounded by C k + (cid:96) −( k + ) .Such a function may be obtained by choosing a smooth function ψ ∶ R d → R satisfying that ψ ≡ B ( ) , ψ ≡ B ( ) (where B ( r ) ∶= { x ∈ R d ∶ ∥ x ∥ ∞ ≤ r } ) andthen setting θ ( v ) ∶= πψ ( v / (cid:96) ) for v ∈ Z d . ● Instead of the expression (8.14) one notes that a discrete Taylor expansion may be performedto write θ w = θ v + P k,v ( w − v ) + ε k,v,w where P k,v is a polynomial of degree at most k and then UANTITATIVE DISORDER EFFECTS IN LOW-DIMENSIONAL SPIN SYSTEMS 45 the higher-order symmetry of the Hamiltonian allows to replace the expression θ w − θ v on theright-hand side of (8.14) by ε k,v,w which is of order at most C k + (cid:96) −( k + ) by our assumptionson θ .The reduced surface tension allows to push the Imry–Ma phenomenon to higher dimensions.Specifically, spin systems with a finite-range smooth Hamiltonian enjoying a higher-ordersymmetry of order k will lose their low-temperature ordered phase in all dimensions d ≤ ( k + ) .Moreover, the strategy used in this paper to obtain a quantitative decay rate can be followedverbatim to yield that (at h = RRRRRRRRRRRR E ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ⟨ σ v ⟩ τ, L ⎤⎥⎥⎥⎥⎦RRRRRRRRRRRR ≤ CL − ( k + )+ d / in dimensions d < ( k + ) . It is further possible that the strategy used in the proof of Theorem 4can be adapted to yield a bound in dimension d = ( k + ) . Comparison with the results of Aizenman–Wehr:
The seminal result of Aizenman–Wehr [5, 6] establishes rigorously the rounding of the first-order phase transitions of low-dimensional spin systems upon the addition of a quenched disorder. Our work presents aquantified version of the Aizenman–Wehr theorem, but applies in somewhat different generalitythan the original result. While we expect that the techniques developed in this work may beextended to a more general setup, closer to that of [5, 6], we have not pursued this direction. Inthis section, we elaborate on the various assumptions made: ● Translation-invariance of the systems:
While the proof of Aizenman and Wehr requires towork in a translation-invariant setup, the techniques developed in this article apply to spinsystems which do not satisfy this assumption. ● Distribution of the disorder:
The result of Aizenman–Wehr applies to a wide class of disorderdistributions while our result is presented only for the Gaussian case. ● Range of the interaction:
The results of [5, 6] apply also to disordered systems with long-rangeinteractions (in which case one-dimensional systems are also of interest) as long as these decayat a sufficiently fast rate. Our results for general two-dimensional disordered spin systems(Section 2) allow the base system to have arbitrary interactions as long as the boundedboundary effect condition (2.2) holds. However, we have opted to restrict to finite-rangedependencies in the noised observables ( f v ) v ∈ Z d (Section 2.1). ● Uniformity of the results in the temperature and external field:
The results of Aizenman–Wehrapply not only for a fixed value of the temperature and external magnetic field (the latter isincorporated into the models there) but also uniformly when these parameters are themselvesallowed to depend on the disorder η . This uniformity shows that there cannot be deviationsfrom the proven behavior at random critical points. In comparison, our results are statedonly for a fixed value of the temperature and, in the general two-dimensional setup, withoutan external magnetic field (though one can be included in the base Hamiltonian). Still, auniform version of our results may be obtained with minor modifications of the proof, asindicated in Remark 7.6. ● Systems with continuous symmetry:
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