aa r X i v : . [ m a t h - ph ] J un Replica Condensation and Tree Decay
Arthur Jaffe ∗ and David MoserHarvard UniversityCambridge, MA 02138, USA Arthur Jaff[email protected], [email protected]
December 18, 2018
Abstract
We give an intuitive method—using local, cyclic replicasymmetry—to isolate exponential tree decay in truncated (connected)correlations. We give an expansion and use the symmetry to showthat all terms vanish, except those displaying replica condensation .The condensation property ensures exponential tree decay.We illustrate our method in a low-temperature Ising system, butexpect that one can use a similar method in other random field andquantum field problems. While considering the illustration, we provean elementary upper bound on the entropy of random lattice surfaces.
Contents
I Introduction 2
I.1 The Ising Model as Illustration . . . . . . . . . . . . . . . . . 4
II The Correspondence Z d ↔ R d III.1 Replica Variables . . . . . . . . . . . . . . . . . . . . . . . . . 6III.2 The Global Replica Group . . . . . . . . . . . . . . . . . . . 6 ∗ The authors thank an anonymous donor, whose gift enabled this collaboration. Arthur Jaffe and David Moser
III.3 The Local Cyclic Replica Group . . . . . . . . . . . . . . . . 7III.4 Irreducible Representations . . . . . . . . . . . . . . . . . . . 7III.5 Replica Boundary Conditions . . . . . . . . . . . . . . . . . . 8III.6 Replica Symmetry is Global, not Local . . . . . . . . . . . . . 9
IV Expectations 10
IV.1 Truncated Expectations . . . . . . . . . . . . . . . . . . . . . 11IV.2 Truncated Functions as Replica Expectations . . . . . . . . . 12
V Replica Condensation 14
V.1 Continents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14V.2 Local Cyclic Replica Symmetry . . . . . . . . . . . . . . . . . 15V.3 Symmetry Ensures Condensation . . . . . . . . . . . . . . . . 16
VI Contours and the Energy 17
VI.1 Contours for Vector Spins ~σ . . . . . . . . . . . . . . . . . . . 17VI.2 Replica Continent Contours . . . . . . . . . . . . . . . . . . . 19 VII Counting Random Surfaces in R d VIII.1 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . 22VIII.2 Details of the Proof . . . . . . . . . . . . . . . . . . . . . . . 23
I Introduction
Symmetry is used widely in physics to unify laws or simplify results. Globalsymmetries often arise and are characterized by Lie groups or their repre-sentation acting on a manifold. Some symmetries, such as gauge symmetry,are local; they are characterized by the action of a group on a bundle overa manifold. Global replica symmetry has been introduced as a symmetry ofthe Hamiltonian of certain interacting systems such as Ising models, randomfields, and quantum fields, leading to valuable insights.In § III we study local replica symmetry. This is not a symmetry of theHamiltonian in general, but it is a symmetry within certain spin configu-rations. This enables us to simplify our expansion of certain expectationsin the low-temperature Ising system in order to exhibit a desired property:exponential tree decay of truncated correlations. This low-temperature ex- eplica Condensation and Tree Decay h σ i σ i · · · σ i n i T , defined in § IV.1.The Ising spins σ i are maps from the unit lattice Z d in d ≥ ±
1. The Hamiltonian is H = k∇ σ k , and the Gibbs factor is e − βH , where β denotes the inverse temperature. We show in § VIII that there are constants a, b such that for δ n = β − b ln n ≥ (cid:12)(cid:12)(cid:12) h σ i σ i · · · σ i n i T (cid:12)(cid:12)(cid:12) ≤ a n n e − δ n τ ( i ,...,i n ) , (I.1)where τ ( i , . . . , i n ) is the length of the minimal tree connecting the n points i , . . . , i n . Note the condition δ n ≥ β ≥ β n , where β n grows atleast as fast as O (ln n ). It would be of interest to eliminate the n -dependencefrom the minimum value of β .Our method uses replica variables, comprising n identical, independentcopies of the original system; one considers expectations in the replicatedsystem that are product expectations for the individual systems. Replicasymmetry is the symmetry of these expectations under a permutation ofthe copies. For a system in a finite volume Λ, with i , . . . , i n ∈ Λ, thesame estimate holds uniformly in Λ. Our method requires unbroken replicasymmetry, so one must impose the same boundary conditions in each replicacopy.We develop a low-temperature expansion, based on the intuitive ideathat individual terms with less than the desired exponential tree-graph decaysum to zero (vanish) due to symmetry under the local cyclic replica group. In § VIII we define and establish convergence of this expansion. The terms in theexpansion are parameterized by replica continents . These replica continentsare bounded by random surfaces . The convergence of our expansion relieson an interplay between energy and entropy estimates; in particular we giveentropy estimates bounding the number of random surfaces that occur inour expansion, as well as energy estimates showing that large islands aresuppressed at a desired rate.Key to our method is the use of local cyclic replica symmetry, to showthat all non-zero terms in our expansion display replica condensation , definedin § V. By this we mean that all the lattice sites i , . . . , i n must live on a singlecontinent. The size of the boundary of the continent must therefore be largerthan τ ( i , . . . , i n ); this is the source of the exponential tree decay. Arthur Jaffe and David Moser
I.1 The Ising Model as Illustration
The Ising system is the simplest example of a statistical mechanics interac-tion. We present our method for such a model on a unit cubic lattice Z d ,with d ≥
2, although our methods clearly apply in more generality. The IsingHamiltonian in volume Λ ⊂ Z d is H Λ = H Λ ( σ ) = 12 k∇ σ k ℓ (Λ) = X nn ∈ Λ (1 − σ i σ j ) , (I.2)where nn denotes the sum over nearest-neighbor pairs of sites in the lattice,namely sites with | i − j | = 1. The partition function Z Λ ,β = X σii ∈ Λ e − βH Λ ( σ ) (I.3)normalizes statistical averages h f i Λ ,β of a function f , namely h f i Λ ,β = 1 Z Λ ,β X σii ∈ Λ f ( σ ) e − βH Λ ( σ ) . (I.4)Often f is a monomial in spins, f = σ i σ i · · · σ i n . The expectation h · i Λ ,β islinear, so one can express the expectation of a general f as a limit of finitelinear combinations of expectations of the form h σ i σ i · · · σ i n i Λ ,β . II The Correspondence Z d ↔ R d Each subset X ⊂ Z d of sites in the lattice Z d can be identified with a subset X ⊂ R d . Define the latter as the union of closed, unit d -cubes (cid:3) i centeredat the lattice sites i ∈ X , as we illustrate in the upper part of Figure 1. Connectedness:
We say that X ⊂ Z d is connected if any two sites in X can be connected by a continuous path through nearest-neighbor lattice sitesin the set X . This agrees with the notion that the interior of the set X ⊂ R d is connected in the ordinary sense. Two cubes are connected if they share a( d − ≤ ( d − eplica Condensation and Tree Decay Z R ←→ ∂ y ∂ y Figure 1: An example for the correspondence between subsets of Z d and R d ,and their boundaries. Boundary:
The boundary ∂X ⊂ R d allows us to define the set ∂X ⊂ Z d ofboundary lattice sites. These boundary sites ∂X ∈ Z d are those lattice sitesin X lying in cubes that share a ( d − ∂X ⊂ R d .By | ∂X | we always refer to the area of the ( d − R d and not the number of points in Z d . (A single cube (cid:3) i , for example,contains exactly 1 boundary lattice site, while | (cid:3) i | = 2 d .) In most instanceswe will call this area the “length” of the boundary, but in some cases wewill also call it the number of faces of the boundary surface. We illustratethe correspondence between the boundary lattice sites and the boundary ofregions in R d in the lower part of Figure 1. Surface:
More generally let a face in R d denote a ( d − d -cubes in R d . A surface Y is a union of( d − | Y | is the number of ( d − Y . Latticesites in Y may lie on either side of the surface Y , but could be limited byselecting an orientation to appropriate sets of faces in Y . Arthur Jaffe and David Moser
Connected Surface:
Define two faces to be adjacent, if they share a ( d − Y to be connected if any two faces in Y can bereached by a continuous path through a sequence of adjacent faces in Y . III Replica Variables and Symmetry
Choose n ∈ Z + and consider n independent copies of a statistical-mechanicalor quantum-field system; these are called n replicas. One can study the prop-erties of expectations under the group of permutations of the replica variables(the replica group ). The n -element subgroup of cyclic permutation of all thecopies is abelian, and it provides useful one-dimensional representations ofreplica symmetry. III.1 Replica Variables
We assume that the different replicas are identical and independent. Theyare defined on the same lattice, they have the same form of interaction, theyare given identical boundary conditions, etc. We label the spin variable atthe lattice site i by σ ( α ) i , where α = 1 , , . . . , n denotes the index of the copy.We also consider the replica spins at site i as a vector ~σ i with the vectorcomponents σ ( α ) i . III.2 The Global Replica Group
The global replica group is the symmetric group S n comprising elements π ∈ S n with action, π : (1 , . . . , n ) ( π , . . . , π n ) . (III.1)The element π ∈ S n acts on the spins, giving a unitary representation, σ ( α ) i ( πσ i ) ( α ) = σ ( π − α ) i , for α = 1 , . . . , n , and for all i . (III.2)The global cyclic replica group S cn is the subgroup of cyclic permutations of n objects, and is generated by the permutation π , π : (1 , . . . , n ) (2 , . . . , n, . (III.3)Treating the indices α modulo n , substitute α = n for α = 0 and write σ ( α ) i (cid:0) π σ i (cid:1) ( α ) = σ ( α − i , for α = 1 , . . . , n , and for all i . (III.4) eplica Condensation and Tree Decay ~σ i π ~σ i , where (cid:0) π ~σ i (cid:1) ( α ) = n X α ′ =1 (cid:0) π (cid:1) α α ′ σ ( α ′ ) i , and (cid:0) π (cid:1) αα ′ = δ α − α ′ . (III.5) III.3 The Local Cyclic Replica Group
Let K denote a subset of the lattice Z d . The local cyclic replica group S cn ( K )is a bundle over S cn defined as the action of S cn on the spins in K and theidentity on the complement. This group is generated by π K which has therepresentation on spins, π K ~σ i = (cid:26) π ~σ i , when i ∈ K ~σ i , when i
6∈ K . (III.6) III.4 Irreducible Representations
The cyclic replica group is abelian, so its irreducible representations are onedimensional. We transform from ~σ i to a set of coordinates ~s i = U~σ i to reducethe representation of S cn . In particular, let ω = e πi/n denote the primitive n th root of unity. Define s ( α ) i = 1 n / n X α ′ =1 ω α ( α ′ − σ ( α ′ ) i , for α = 1 , . . . , n . (III.7)Note that for n > s -variables may be complex, even though the original σ -spins are real. The choice (III.7) defines the entries of the matrix U as U αα ′ = n − / ω α ( α ′ − . This is Fourier transform in the replica space. Proposition III.1.
The matrix U is unitary with eigenvalues ω α , for α =1 , . . . , n . Let D be the diagonal matrix with D αα ′ = ω α δ αα ′ . Then π ~s i = D~s i . (III.8) Proof.
For ν an integer (modulo n ), n X α =1 ω − να = n δ ν . (III.9) Arthur Jaffe and David Moser
Thus (
U U ∗ ) αα ′ = n X β =1 U αβ U α ′ β = 1 n n X β =1 ω ( α − α ′ )( β − = δ αα ′ . (III.10)Since π acts on the ~σ i components according to (III.4), this means that (cid:0) π ~s i (cid:1) ( α ) = ω α ( ~s i ) ( α ) = n X α ′ =1 D αα ′ ( ~s i ) ( α ) , (III.11)which is (III.8).The inverse change of coordinates is σ ( γ ) i = 1 n / n X α =1 ω − ( γ − α s ( α ) i , for γ = 1 , . . . , n . (III.12)A further corollary of the unitarity of U is the fact that for any i, j n X α =1 σ ( α ) i σ ( α ) j = h ~σ i , ~σ j i ℓ = h U~σ i , U~σ j i ℓ = h ~s i , ~s j i ℓ = n X α =1 s ( α ) i s ( α ) j . (III.13)In particular, the expression on the right side of this identity is always real.Furthermore, each individual term on the right is invariant under the ele-ments of the local, cyclic replica group S cn ( K ) as long as both i, j ∈ K orboth i, j
6∈ K . III.5 Replica Boundary Conditions
We consider finite volume Hamiltonians that, along with their boundaryconditions, have the global replica group as a symmetry. If one wished toinvestigate the breaking of the replica group in the infinite volume limit, thenone might explicitly break replica symmetry in a finite volume by imposingdifferent boundary conditions for different replica copies of the system.Since our system is originally given in terms of the variables σ i , onedescribes the boundary conditions in the volume Λ in terms of the variables σ i for i ∈ ∂ Λ, with ∂ Λ defined in § II. eplica Condensation and Tree Decay σ ( α ) i = σ i , for all i ∈ ∂ Λ , and all α = 1 , . . . , n . (III.14)In order to simplify the discussion, we impose +1 boundary conditions ineach replica copy: set ~σ i = (+1 , . . . , +1) , when i ∈ ∂ Λ . (III.15)The resulting boundary conditons for ~s are ~s i = (cid:0) , , . . . , , n / (cid:1) , when i ∈ ∂ Λ . (III.16) III.6 Replica Symmetry is Global, not Local
Define the total replica Hamiltonian H replica as the sum of the Hamiltoniansfor the replica copies of the Hamiltonian in volume Λ, H replica = H replica ( ~σ ) = 12 k∇ ~σ k ℓ (Λ) = 12 n X α =1 X nn ∈ Λ (cid:16) σ ( α ) i − σ ( α ) j (cid:17) . (III.17) Proposition III.2.
Consider the replica Hamiltonian (III.17) .i. As a function of the variables ~s , one has H replica = 12 k∇ ~σ k ℓ (Λ) = 12 k∇ ~s k ℓ (Λ) = 12 n X α =1 X nn ∈ Λ (cid:12)(cid:12)(cid:12) s ( α ) i − s ( α ) j (cid:12)(cid:12)(cid:12) . (III.18) ii. The replica Hamiltonian (III.18) is invariant under a global replicapermutation π ∈ S n defined in (III.2) , namely H replica ( π~s ) = H replica ( ~s ) . (III.19) iii. In general, the replica Hamiltonian is not invariant under the localcyclic replica group S cn ( K ) defined in (III.6) . Arthur Jaffe and David Moser
Proof.
The relation (III.13) shows that H replica has the form (III.18). Theinvariance under the global replica group follows by considering the effect on H replica expressed in the ~σ variables, where the transformation permutes thevarious terms H Λ ( σ ( α ) ) in the first expression for H replica in (III.17).In order to see that H replica ( ~σ ) is not invariant under the local cyclic replicagroup, we give a configuration ~σ and set K that provides a counterexamplein the case n = 2. It is easiest to visualize this configuration by illustratingit; see the left side of Figure 2. We choose K to be the centermost square inthe configuration (with σ (1) = +1 and σ (2) = − π K ∈ S cn ( K )to flip the spins in K . The action of π K produces the configuration on theright side of the figure, and it lowers the energy by 4 | ∂ K| . In other words, H replica ( ~σ ) − H replica ( π K ~σ ) = 4 | ∂ K| , showing that H replica is not invariant underthe action of S cn ( K ). σ (1) :PSfrag replacements ++ − −→ PSfrag replacements + − σ (2) :PSfrag replacements + − −→ PSfrag replacements + − Figure 2: A counter-example to local cyclic replica symmetry.
IV Expectations
Define the expectation ≪ · ≫ Λ ,β for the replicated system as follows: for afunction F ( ~σ ), let ≪ F ≫ Λ ,β = 1 Z X ~σii ∈ Λ F ( ~σ ) e − βH replica ( ~σ ) , (IV.1)where Z = Z n , with Z is given in (I.3). In case that F ( ~σ ) = f ( σ ( α ) ) onlydepends on one component σ ( α ) , the expectation ≪ · ≫ Λ ,β reduces to the eplica Condensation and Tree Decay h · i Λ ,β . In this case ≪ f ( σ ( α ) ) ≫ Λ ,β = h f ( σ ) i Λ ,β , for α = 1 , . . . , n . (IV.2)We now introduce the generating function S ( µ ) for expectations of prod-ucts of spins. Let µ be a function from Λ to C and let σ ( µ ) = X i ∈ Λ µ i σ i , and correspondingly σ ( α ) ( µ ) = X i ∈ Λ µ i σ ( α ) i . (IV.3)Then define S ( µ ) = (cid:10) e σ ( µ ) (cid:11) Λ ,β = ≪ e σ ( α ) ( µ ) ≫ Λ ,β . (IV.4)The expectations of n spins are derivatives of the generating function, h σ i σ i · · · σ i n i Λ ,β = ∂ n ∂µ i ∂µ i · · · ∂µ i n S ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) µ i =0 = ≪ σ (1) i σ (1) i · · · σ (1) i n ≫ Λ ,β . (IV.5)The expectations (IV.5) are n -multi-linear, symmetric, functions of the spins, h σ ( µ ) n ) i Λ ,β = n X i ,...,i n =1 µ i · · · µ i n h σ i σ i · · · σ i n i Λ ,β . (IV.6)One can recover the expectation h σ i σ i · · · σ i n i Λ ,β from the expectations ofpowers of σ ( µ ) by polarization, h σ i σ i · · · σ i n i Λ ,β = 12 n n ! X ǫ ,...,ǫ n = ± ǫ · · · ǫ n h ( ǫ σ i + · · · + ǫ n σ i n ) n i Λ ,β . (IV.7) IV.1 Truncated Expectations
The truncated expectation of a product of n spins is a generalization of thecorrelation of two spins. The truncated expectation vanishes asymptoticallyas one translates any subset of the spin locations a large distance away fromthe others.The generating function of the connected expectations is G ( µ ) = ln S ( µ ) = ln (cid:10) e σ ( µ ) (cid:11) Λ ,β . (IV.8)2 Arthur Jaffe and David Moser
One defines the truncated (connected) expectations as h σ i σ i σ i · · · σ i n i TΛ ,β = ∂ n ∂µ i ∂µ i · · · ∂µ i n G ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) µ i =0 . (IV.9)A standard representation of h σ i σ i σ i · · · σ i n i TΛ ,β in terms of sums of prod-ucts of expectations can be formulated in terms of the set P of partitions of { i , i , . . . , i n } . Suppose that a set P ∈ P has cardinality | P | . Then h σ i σ i σ i · · · σ i n i Λ ,β = X P Y P ∈P (cid:10) σ P (cid:11) TΛ ,β . (IV.10)Like the expectations (IV.5), the n -truncated expectations satisfy the n -multi-linear relation (IV.6)–(IV.7). Thus h σ ( µ ) n i TΛ ,β = n X i ,...,i n =1 µ i · · · µ i n h σ i σ i σ i · · · σ i n i TΛ ,β , (IV.11)and h σ i σ i · · · σ i n i TΛ ,β = 12 n n ! X ǫ ,...,ǫ n = ± ǫ · · · ǫ n h ( ǫ σ i + · · · + ǫ n σ i n ) n i TΛ ,β . (IV.12) IV.2 Truncated Functions as Replica Expectations
The form of the replica variables ~s leads to an elementary representationof the truncated (connected) expectations of products of spins. Ultimatelywe show that this yields exponential decay at low temperatures with a rategoverned by the length of the shorted tree-graph connecting all the spins. (Asimilar argument presumably works at high temperature.)Our expansion method uses replica symmetry to arrange that each termin the expansion either exhibits the desired decay rate, or else it is canceled byother terms as a consequence of local cyclic replica symmetry. We begin byestablishing a known representation of the connected correlation of n spins asan expectation of n replica variables introduced above. This representationwas discovered by P. Cartier (unpublished); our presentation is based onSylvester’s treatment [2] using s (1) . Let g . c . d . denote the greatest commondivisor. eplica Condensation and Tree Decay Proposition IV.1.
Let ~s be defined in (III.7) with n replica copies, and let γ ∈ (1 , . . . , n ) satisfy g . c . d . ( n, γ ) = 1 . Then h σ i σ i · · · σ i n i TΛ ,β = n ( n − / ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ Λ ,β . (IV.13) Lemma IV.2.
For all γ = 1 , . . . , n , ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ TΛ ,β = n − ( n − / h σ i σ i · · · σ i n i TΛ ,β . (IV.14) Proof.
Using the multi-linearity (IV.11), and its analog for the expectations h · i Λ ,β and ≪ · ≫ Λ ,β of the truncated functions, we infer that ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ TΛ ,β = n − n/ ≪ n X α ,...,α n =1 ω γα + ··· + γα n − γn σ ( α ) i σ ( α ) i · · · σ ( α n ) i n ≫ TΛ ,β = n − n/ n X α ,...,α n =1 ω γα + ··· + γα n − γn ≪ σ ( α ) i σ ( α ) i · · · σ ( α n ) i n ≫ TΛ ,β . (IV.15)Since the different components of ~σ i are independent, the expectations on theright vanishes unless α = · · · = α n . In this case the truncated expectationof each copy equals the truncated expectation of the original spins, and thesum yields n such terms. Therefore (IV.14) holds as claimed. Lemma IV.3.
Let kγ = 0 (modulo n ) . Then ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i k ≫ Λ ,β = 0 . (IV.16) Proof.
Expand the expectation ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i k ≫ Λ ,β = 1 n n/ n X α ,...,α n =1 ω γα + ··· + γα k − γk ≪ σ ( α ) i σ ( α ) i · · · σ ( α n ) i k ≫ Λ ,β = 1 n n/ n X α ,...,α n =1 ω γα + ··· + γα k − γk ≪ σ ( α − i σ ( α − i · · · σ ( α n − i k ≫ Λ ,β . (IV.17)4 Arthur Jaffe and David Moser
In the second equality, we use the symmetry of the expectation ≪ · ≫ Λ ,β under the global cyclic replica group S cn ∋ π . Therefore ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i k ≫ Λ ,β = ω γk ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i k ≫ Λ ,β . (IV.18)As long as γk = 0 (modulo n ), it is the case that ω γk = 1. Therefore theexpectation must vanish. Proof of the Proposition.
The relation (IV.10) also holds for the replicaexpectations, ≪ s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ Λ ,β = X P Y P ∈P ≪ s ( γ ) P ≫ TΛ ,β . (IV.19)Because g . c . d . ( n, γ ) = 1, it is the case that kγ = 0 (modulo n ) for all k = 1 , . . . , n −
1. Thus we can apply Lemma IV.3 to each such k , and onlythe partition P with all n elements in one set survives in (IV.19). We infer ≪ s ( γ ) i s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ Λ ,β = ≪ s ( γ ) i s ( γ ) i s ( γ ) i · · · s ( γ ) i n ≫ TΛ ,β . (IV.20)Using Lemma IV.2 then completes the proof. V Replica Condensation
In this section we investigate certain classes of configurations ~σ of the replicaspins. We see that for each class of configurations, there is a local cyclicreplica group (see § III.3) under which the Hamiltonian H replica of (III.17) isinvariant. This leads to the phenomenon of replica condensation in whichall the spin localizations i , . . . , i n must be localized within a given region K ⊂
Λ that we call a continent.
V.1 Continents
Each configuration of spins ~σ in the volume Λ defines a sea S ( ~σ ), surroundinga set of continents K ( ~σ ). The sea starts at the boundary boundary ∂ Λ ofthe region Λ. The boundary of a continent appears if any one of the compo-nents of ~σ changes its value. Continents have a substructure arising from the eplica Condensation and Tree Decay σ ( α ) within the conti-nent. We say more about this substructure when defining replica continentcontours in § VI.2. In the following we utilize the notion of “connectedness”introduced in § II.
Definition V.1.
Consider a configuration ~σ . The replica sea S ( ~σ ) is the con-nected component of the set { i | ~σ i = (+1 , . . . , +1) } that meets the boundary ∂ Λ of Λ . The continents K j are the connected components of the comple-mentary set, S c ( ~σ ) = K ∪ · · · ∪ K r . The set of continents K ( ~σ ) is K ( ~σ ) = {K , . . . , K r } . (V.1)We illustrate this definition in Figure 3.PSfrag replacements K K K K K S ( ~σ )Figure 3: The set of continents K ( ~σ ) = {K , . . . , K } in the sea S ( ~σ ). V.2 Local Cyclic Replica Symmetry In § III.6 we saw that a global replica symmetry transformation leaves H replica ( ~σ ) invariant, and that a local replica symmetry transformation doesnot necessarily do so. We now recover local cyclic replica symmetry by choos-ing the localization K in S cn ( K ) to be a continent. Proposition V.2.
Let
K ∈ K ( ~σ ) . Then the local cyclic replica group S cn ( K ) defined in (III.6) preserves the continent K and the Hamiltonian H replica ( ~σ ) .For π K ∈ S cn ( K ) , H replica ( ~σ ) = H replica ( π K ( ~σ )) . (V.2)6 Arthur Jaffe and David Moser
Proof.
The action of S cn ( K ) on ~σ leaves invariant spins ~σ i = (+1 , . . . , +1),so it changes neither the sea S ( ~σ ) nor the definition of continents. Henceit also does not change the contribution of nearest neighbor spins to theenergy either inside or outside the continent. The local permutation also doesnot alter the energy across the island boundary, because all the componentsoutside the island have value +1 and are invariant under the permutation. V.3 Symmetry Ensures Condensation
We now establish the property of condensation. We use the representation(IV.13) for the truncated correlation function of n spins. We may choose any γ with g . c . d . ( n, γ ) = 1, so for simplicity we consider the case γ = 1. Proposition V.3 ( Condensation).
In the expectation ≪ s (1) i · · · s (1) i n ≫ Λ ,β ,any configuration ~σ giving a nonzero contribution has all the sites i , . . . , i n ∈K lying in a single continent K ∈ K ( ~σ ) . Lemma V.4.
Consider a given configuration ~σ and a continent K ∈ K ( ~σ ) containing at least one but not all the sites i , . . . , i n . Let π k K denote π K applied k times. Then n − X k =0 (cid:16) π k K s (1) i (cid:17) · · · (cid:16) π k K s (1) i n (cid:17) e − βH replica ( π k K ( ~σ ) )= n − X k =0 s (1) i (cid:0) π k K ( ~σ ) (cid:1) · · · s (1) i n (cid:0) π k K ( ~σ ) (cid:1) e − βH replica ( π k K ( ~σ ) )= 0 . (V.3) Proof.
From Proposition V.2 we infer that the energy in the permuted con-figuration is unchanged by the permutation, H replica (cid:0) π k K ( ~σ ) (cid:1) = H replica ( ~σ ) . (V.4)Therefore, we only need consider the changes to the spins s (1) i k . Let l = |{ k | i k ∈ K}| denote the number of sites i , . . . , i k that lie in K ; clearly 1 ≤ l < n . According to Proposition III.1, the application of π K to s (1) i gives a eplica Condensation and Tree Decay ω for i ∈ K . The sum equals n − X k =0 ω kl s (1) i s (1) i · · · s (1) i n e − βH replica ( ~σ ) = s (1) i s (1) i · · · s (1) i n e − βH replica ( ~σ ) n − X k =0 ω kl = 0 . (V.5) Proof of Proposition V.3.
The expectation is ≪ s (1) i · · · s (1) i n ≫ Λ ,β = X ~σ s (1) i · · · s (1) i n e − βH replica ( ~σ ) / Z . (V.6)If ~σ is a configuration where some site i k lies in the sea i k ∈ S ( ~σ ) then thespin has the value of the boundary, s (1) i k = 0. We also have s (1) i k = 0, if i k ∈ K and all the σ ( α ) take the same values on K .Therefore, the only contributing configurations have all the sites i k lyingin continents where π K actually yields new configurations. In this case, thesum in Lemma V.4 is a sub-sum of (V.6). According to the lemma the sumis only nonzero if all or none of the i k lie in the contintent K . VI Contours and the Energy
VI.1 Contours for Vector Spins ~σ For each component σ ( α ) of the vector spin, we can define contours in theusual statistical mechanics sense. These contours are the boundaries betweenislands with different values of σ ( α ) , as defined in § II. They are subsets ofthe lattice dual to Z d , consisting of ( d − d -cubes.The ~σ contours are the direct sum of contours in the individual com-ponents. In order to picture the boundaries of ~σ , we assign colors to thedifferent components, corresponding to the label α used above. We illustratethese contours for a particular configuration in the case n = 2 in Figure 4(a)–Figure 4(c).8 Arthur Jaffe and David Moser
PSfrag replacements + − − K K color 1color 2 S ( ~σ ) (a) The contours of σ (1) PSfrag replacements ++ − − K K color 1color 2 S ( ~σ ) (b) The contours of σ (2) PSfrag replacements+ − K K color 1color 2 S ( ~σ ) (c) The contours of ~σ PSfrag replacements+ − K K color 1color 2 S ( ~σ ) (d) The set of continents K ( ~σ ) = {K , K } PSfrag replacements+ − K K color 1color 2 S ( ~σ ) (e) The continent contours ~C ( K , ~σ ) Figure 4: An illustration of contours and continents in the case n = 2. eplica Condensation and Tree Decay VI.2 Replica Continent Contours
Here we define appropriate replica continent contours ~C ( K , ~σ ) in order toanalyze the probability Pr( r ) of the occurrence of configurations containing acontinent with a contour of length r . We do not define ~C as the boundary ∂ K .The problem is: while this boundary is a contour for ~σ , it is not necessarilya contour for a component σ ( α ) . Usually ∂ K consists of segments of contoursof the components.To estimate Pr( r ) we use the relation between the configuration ~σ with thereplica continent contour ~C ( K , ~σ ) and a configuration ~σ ∗ with the contourremoved. This transformation removes all the contours of the componentspins σ ( α ) that contribute to ∂ K . With this motivation, we now give theappropriate construction. Definition VI.1.
For
K ∈ K ( ~σ ) define the replica continent contour of K in the configuration ~σ as the vector ~C ( K , ~σ ) with components C ( α ) ( K , ~σ ) = union of contours C for σ ( α ) with | C ∩ ∂ K| 6 = 0 , (VI.1) where | · | is the measure of ( d − -surfaces. This is the subset of contoursfor ~σ meeting the boundary of the continent ∂ K . See the example in Figure 4(e). In a generic configuration, these con-tours touch the boundary and penetrate arbitrarily into the interior of thecontinent.Several different configurations of the spin ~σ may have different contours,but a common continent K . Define the set of possible contours for the con-tinent K as C ( K ) = n ~C ( K , ~σ ) | where K ∈ K ( ~σ ) o . (VI.2)Finally, the length of any contour ~C ∈ C ( K ) is just the sum over the lengthof the constituent contours, (cid:12)(cid:12)(cid:12) ~C (cid:12)(cid:12)(cid:12) = n X α =1 (cid:12)(cid:12) C ( α ) (cid:12)(cid:12) . (VI.3)With these definitions it is obvious that removing ~C ( K , ~σ ) in the configu-ration ~σ is well-defined. We just remove the respective contours C ( α ) ( K , ~σ )for the components σ ( α ) , by flipping the sign of all the spins inside thesecontours.0 Arthur Jaffe and David Moser
Definition VI.2.
For a configuration ~σ and a continent K ∈ K ( ~σ ) , write ~σ ∗ for the configuration where the contour ~C ( K , ~σ ) for the continent has beenremoved as described above. As a consequence of the removal of the replica continent contour theenergy H replica is decreased by two times the length of the removed contours.This the generalization of the fact that for each component spin, the energyis given by two times the total length of the contours, H replica ( ~σ ∗ ) = H replica ( ~σ ) − (cid:12)(cid:12)(cid:12) ~C ( K , ~σ ) (cid:12)(cid:12)(cid:12) . (VI.4) VII Counting Random Surfaces in R d In order to prove the tree decay we need an exponential bound on the numberof possible connected contours. These are surfaces in R d composed of r faces,each a unit ( d − random surfaces and prove a boundthat holds for general connected unions of faces, as defined in § II. We alsouse the term adjacent faces as in that section, to indicate that two faces sharea ( d − Definition VII.1.
Let N ( r ) denote the number of connected, random sur-faces of dimension ( d − , which contain exactly r faces, including a givenface S . Proposition VII.2.
There is a constant a (independent of dimension) suchthat for k d = a d , N ( r ) ≤ k rd . (VII.1) Proof.
The idea of the proof is to map each connected surface onto a rootedtree-graph, whose edges connect the centers of adjacent faces of the surface,and which touches each face. We say that the graph covers the surface.One then counts the number of possible surfaces that can correspond to onegraph. The product of the number of possible tree graphs, times the numberof surfaces per graph, gives our bound.The tree graph will have length r and r − p of S , see Figure 5. The first branch of the tree connects theroot p to the center p of a face adjacent to S . From there we draw anotheredge connecting to the center of a new adjacent face (but we do not return eplica Condensation and Tree Decay p p p p p p S PSfrag replacements p p p p p p S Figure 5: A surface covered by a tree graph and the corresponding tree graphrooted at the center of S . To simplify the illustration, all the angles betweenthe faces are set to 180 ◦ , while in general they may equal 90 ◦ , 180 ◦ , or 270 ◦ .to p ). If all the adjacent surface elements are already in the tree graph, wecannot continue this branch. At this point we move in the reverse directionalong the branch, face by face, until we reach a face having an adjacent facethat is not yet covered by the tree. Starting at this place we start a newbranch. We continue in this manner until we cover the entire surface.In this manner we assign at least one tree diagram to every connectedsurface. This also means that every possible connected surface with r facescan be constructed by choosing a tree graph and attaching new faces in theorder given by the tree structure. The number of planar tree graphs with r − C r − , see example 6.19.e of Stanley [1].Hence C r − = 1 r (cid:18) r − r − (cid:19) ≤ (2 e ) r , (VII.2)where the bound follows from the elementary inequality (cid:18) vw (cid:19) ≤ (cid:16) evw (cid:17) w . (VII.3)An upper bound for the number of ways to add a single face as one buildsup the surface along the tree-graph is 2 d − ·
3. A face has 2 d − sides to attachan adjacent face, and every attachment can be done with one of the angles90 ◦ , 180 ◦ , or 270 ◦ . Therefore we infer the bound (VII.1) with a = 3 e , namely N ( r ) ≤ (2 d − · r C r − ≤ (2 d − · · e ) r = k rd . (VII.4)2 Arthur Jaffe and David Moser
VIII Tree Decay
In this section we prove the decay bound for the truncated correlation func-tions. We base the proof on condensation. Starting from the representation(IV.13), namely ≪ s (1) i · · · s (1) i n ≫ Λ ,β = n − ( n − / h σ i · · · σ i n i TΛ ,β , (VIII.1)we use the fact established in Proposition V.3 that every non-vanishing con-tribution contains a continent K with all the points i , . . . , i n . Proposition VIII.1.
There are constants a, b depending on d , but indepen-dent of Λ , such that if ≤ δ = β − b ln n (hence requiring β ≥ β n = O (ln n ) ),then the truncated correlation functions satisfy (cid:12)(cid:12) h σ i · · · σ i n i TΛ ,β (cid:12)(cid:12) ≤ a n n e − δτ ( i ,...,i n ) . (VIII.2) Here τ ( i , . . . , i n ) is the length of the shortest tree connecting i , . . . , i n . VIII.1 Outline of the Proof
We have shown in Proposition V.3 that each non-vanishing contribution tothe expectation (VIII.1) contains a condensate continent K containing all thepoints i , . . . , i n . As a consequence, every possible replica contour ~C ∈ C ( K )has minimal length τ ( i , . . . , i n ).We formulate the sum over configurations h σ i · · · σ i n i TΛ ,β = n ( n − / Z X ~σ s i · · · s i n e − βH replica ( ~σ ) , (VIII.3)as a sum over configurations with contours ~C of length r and a sum over r . We claim that the probability Pr( r ) that a replica contour ~C occurs with (cid:12)(cid:12)(cid:12) ~C (cid:12)(cid:12)(cid:12) = r satisfies the boundPr( r ) ≤ e − β | ~C | = e − βr . (VIII.4) eplica Condensation and Tree Decay ~C . These estimates, together with the fact that (cid:12)(cid:12)(cid:12) s (1) i (cid:12)(cid:12)(cid:12) ≤ n / , yield the desiredbound. We now break the proof into a sequence of elementary steps. VIII.2 Details of the Proof
Rewrite the Sum:
Consider the sum (VIII.3), with the restriction ofProposition V.3. Recall that the replica continent borders ~C = ~C ( K , ~σ ), andthe set of configurations containing such a replica continent C ( K ) ∋ ~C ( K , ~σ )is given in Definition VI.1. One can rewrite the sum as an iterated sum, X ~σ = ∞ X r = τ ( i ,...,i n ) ′ X K , ~C ′′ X ~σ . (VIII.5)For fixed K and ~C , the sum P ′′ denotes the sum over configurations con-taining the continent K ∈ K ( ~σ ) with the continent border ~C = ~C ( K , ~σ ), ′′ X ~σ = X ~σ with K ∈ K ( ~σ ) , ~C = ~C ( K , ~σ ) . (VIII.6)The sum P ′ ranges over the possible continents K containing the n sites i , . . . , i n , and their possible borders ~C of length (cid:12)(cid:12)(cid:12) ~C (cid:12)(cid:12)(cid:12) = r . Thus ′ X K , ~C = X K⊃{ i ,...,i n } X ~C ∈ C ( K )with ˛˛˛ ~C ˛˛˛ = r . (VIII.7)Finally we sum over r , which is bounded from below by the minimal size τ ( i , . . . , i n ).One interprets the sum P ′′ as the energy contribution to the sum, namelythe probability Pr( r ) = 1 Z ′′ X e − βH ( ~σ ) , (VIII.8)4 Arthur Jaffe and David Moser for the states ~σ with K ∈ K ( ~σ ). Likewise one interprets the sum P ′ as the entropy contribution to the sum. Define the entropy factor N ( r ) by N ( r ) = ′ X K , ~C . (VIII.9)The entropy counts the number of different shapes for ~C .Using | σ i | = 1, one has (cid:12)(cid:12)(cid:12) s (1) i (cid:12)(cid:12)(cid:12) ≤ n / . Thus we obtain the bound (cid:12)(cid:12) h σ i · · · σ i n i TΛ ,β (cid:12)(cid:12) = n ( n − / (cid:12)(cid:12)(cid:12) ≪ s (1) i · · · s (1) i n ≫ Λ ,β (cid:12)(cid:12)(cid:12) ≤ n ( n − / Z ∞ X r = τ ′ X K , ~C ′′ X ~σ (cid:12)(cid:12)(cid:12) s (1) i · · · s (1) i n (cid:12)(cid:12)(cid:12) e − βH ( ~σ ) ≤ n ( n − / ∞ X r = τ n n/ N ( r ) Pr( r )= n n − ∞ X r = τ N ( r ) Pr( r ) . (VIII.10)In the following we prove bounds on Pr( r ) and on N ( r ) that depend only on r , on β , and on the dimension d . Bound the Entropy:
We show that there are constants
A, B dependingonly on d such that N ( r ) satisfies the exponential bound, N ( r ) ≤ AB r n r . (VIII.11)We obtain this result by constructing the border contour ∂ K and at-taching l colored sub-contours. In this way one constructs any possible ~C satisfying the conditions above. The geometry of the contour (which mustsurround i ) requires that the starting face we choose in constructing ~C mustlie in a cube of side-length ( r − i . Such a cube contains atmost dr d possible starting faces.Using Proposition VII.2, the number of possible border contours is lessthan dr d N ( r ) ≤ dr d k rd . We now build up the full contour ~C by attachingat least one and at most r subcontours to ∂ K to obtain the total number eplica Condensation and Tree Decay r . This can be done in a number of ways. For l sub-contours, thenumber of ways is bounded by the product of combinatorial factors: r l for the starting faces on ∂ K , k rd for the shapes, n l for the colors, (cid:0) r − l − (cid:1) for the lengths,1 /l ! as the ordering of the subcontours is irrelevant.Therefore N ( r ) ≤ dr d k rd r X l =1 r l k rd n l (cid:18) r − l − (cid:19) l ! . (VIII.12)We use the elementary inequalities r d ≤ d ! e r , and (cid:18) r − l − (cid:19) ≤ r . (VIII.13)Then N ( r ) ≤ dr d k rd n r r X l =1 r l l ! (cid:18) r − l − (cid:19) ≤ dd ! (cid:0) e k d (cid:1) r n r . (VIII.14)This bound has the form (VIII.11) with A = dd ! and B = 2 e k d . Bound the Energy Factor:
The energy bound has the formPr( r ) ≤ e − βr , (VIII.15)where K (implicitely contained in P ′ ) is any fixed connected set with { i , . . . , i n } ⊂ K and ~C ∈ C ( K ) is any fixed extended border with (cid:12)(cid:12)(cid:12) ~C (cid:12)(cid:12)(cid:12) = r .The idea is to compare every summand in the numerator to a summand inthe denominator. For any given ~σ with K ∈ K ( ~σ ) and ~C ( K , ~σ ) = ~C , wecan take away the contours in ~C obtaining the unique ~σ ∗ as described inDefinition VI.2. Because of the difference in energy this gives an additionalfactor e − βr for the term in the numerator. As the procedure works for all thesummands, we inferPr( r ) = P ′′ ~σ e − βH ( ~σ ) P ~σ e − βH ( ~σ ) ≤ P ′′ ~σ e − βH ( ~σ ∗ ) e − βr P ′′ ~σ e − βH ( ~σ ∗ ) = e − βr . (VIII.16)6 Arthur Jaffe and David Moser
Tree Decay:
The bound (VIII.2) now follows. Using (VIII.10), one has (cid:12)(cid:12) h σ i · · · σ i n i TΛ ,β (cid:12)(cid:12) ≤ n n − ∞ X r = τ N ( r ) Pr( r ) ≤ n n − ∞ X r = τ AB r n r e − βr = n n − A ∞ X r = τ e − ( β − b ln n ) r , (VIII.17)where b = ln B and where τ = τ ( i , . . . , i n ). The last sum converges for β > b ln n . With 1 ≤ δ n = β − b ln n , this gives (cid:12)(cid:12) h σ i · · · σ i n i TΛ ,β (cid:12)(cid:12) ≤ n n − A (1 − e − δ n ) − e − δ n τ . (VIII.18)As 1 ≤ δ n , one can take a = A (1 − e − δ n ) − ≤ Ae ( e − − . (VIII.19)This completes the proof of Proposition VIII.1. (cid:3) References [1] Richard Stanley.
Enumerative Combinatorics, Volume 2 . CambridgeUniversity Press. Cambridge Studies in Advanced Mathematics 62, Cam-bridge, New York 1999.[2] Garrett Sylvester. Representations and inequalities for Ising model Ursellfunctions.