Absence of replica symmetry breaking in the Edwards-Anderson model near zero temperature
aa r X i v : . [ m a t h - ph ] F e b Absence of replica symmetry breakingin the Edwards-Anderson model near zero temperature
C. Itoi , H. Shimajiri and Y. Sakamoto , Department of Physics, GS & CST, Nihon University,Kandasurugadai, Chiyoda, Tokyo 101-8308, Japan Laboratory of Physics, CST, Nihon University,Narashinodai, Funabashi-city, Chiba 274-8501, JapanFebruary 9, 2021
Abstract
It is proven that the ground state is unique in the Edwards-Anderson model for almost allcontinuous random exchange interactions, and any excited state with the overlap less thanits maximal value has large energy in dimensions higher than two with probability one. Sincethe spin overlap is shown to be concentrated at its maximal value in the ground state, replicasymmetry breaking does not occur in the Edwards-Anderson model near zero temperature.
The replica symmetry breaking (RSB) in mean field spin glass models have been studied ex-tensively, since Talagrand proved the Parisi formula [17] for the free energy density of theSherrington-Kirkpatrick (SK) model [19] rigorously [21, 20]. Recently, Auffinger, Chen andZeng prove that there exist infinitely many spin confiugurations having infinitesimal energy gapand any value of spin overlap in the SK model with probability one in the Parisi measure [2]. Onthe other hand, there have been many studies also for short range spin glass models, such as theEdwards-Anderson (EA) model [6]. If critical phenomena in short range spin model is identicalto those in mean field spin model in higher than an upper critical dimension as believed, theEA model has RSB phase in higher dimensions. It is quite important question whether the EAmodel has RSB phase in some higher dimension or not. There are two conflicting ideas ‘theRSB picture’ by Parisi [18] and ‘the droplet picture’ by Fisher and Huse [7] to understand thelow temperature behavior of short range spin glass models, such as the EA model. Newmanand Stein have supported the droplet picture that a short range spin glass model should have apure Gibbs state, then the RSB picture is unnatural [13]. Although this question in statisticalphysics has been discussed for four decades after the discovery of the Parisi formula [17] forthe SK model, there has never been any clear answer for the EA model. There are severalrigorous results for RSB in low temperature region of short range disordered spin models. Nishi-mori and Sherrington have shown that the overlap is concentrated at its expectation value onthe Nishimori line located out of the spin glass phase in the EA model [14, 15]. Arguin andDamron have proven that the number of ground states in the EA model is either 2 or ∞ withprobability one on the half-plane [4]. Recently, Chatterjee has proven that the random fieldIsing model has no extended RSB phase in any dimension [5]. This theorem is proven usingthe Fortuin-Kasteleyn-Ginibre (FKG) inequality [8], the Ghirlanda-Guerra identities[1, 9] andthe Chatterjee inequalities [5]. While the Ghirlanda-Guerra identities are well-known to holduniversally in spin systems with Gaussian random interactions, the FKG inequality is valid onlyin the random field Ising model with positive definite exchange interactions [8]. Quite recently,1t is proven that the ground state in the EA model is unique in any dimension for almost allcontinuous random exchange interactions under a condition that a single spin breaks the global Z symmetry [12]. It is also proven that RSB does not occur in the EA model in any finitedimensions at zero temperature.In the present paper, it is proven that the existence of a low energy state with the overlap lessthan its maximal value is rare event in dimensions higher than two. This implies that RSB doesnot occur in the EA model in any finite dimensions in sufficiently low temperature. Uniquenessof the ground state and the absence of RSB in the EA model near zero temperature confirmsthe droplet picture. Behaviors of short range spin glass models turn out to be much differentfrom those of mean field spin glass models. The present paper is organized as follows. In section2, the proof in Ref. [12] for uniqueness of ground state in the EA model is reviewed. In section3, it is proven that any excited state above the unique ground state with the overlap less thanits maximal value has large energy in dimensions higher than two with probability one. Thistheorem implies that the droplet picture is the correct picture to understand behaviors of theEA model in sufficiently low temperature. Consider d -dimensional hyper cubic lattice Λ L = Z d ∩ ( − L/ , L/ d with an even integer L > L containts L d sites. Define a set of nearest neighbor bonds by B Λ = {{ i, j } ∈ Λ L || i − j | = 1 } . Note | B Λ | = | Λ L | d. Let Σ Λ := {− , } Λ L be a set of spin configurations σ : Λ L → {− , } . Abond spin σ b denotes σ b = σ i σ j for a bond b = { i, j } ∈ B Λ . Let g = ( g b ) b ∈ B Λ be a sequence of independent and identicallydistributed (i.i.d) random variables. Let ρ ( g b ) be a probability density function given by P ( g ≤ g b ≤ g ) = Z g g ρ ( g b ) dg b . (1) E denotes the expectation value of a function f ( g ) of sequence g = ( g b ) b ∈ B L E f ( g ) := Z Y b ∈ B L dg b ρ ( g b ) f ( g ) . The expectation of g b and variance are given by E g b = g , E ( g b − g ) = 1 , (2)for g ∈ R . Denote J = J g . The Hamiltonian of this model H Λ ( σ, J, g ) = − X b ∈ B Λ J g b σ b , (3)is a function of spin configuration σ ∈ Σ Λ and a random sequence g . This Hamiltonian isinvariant under the action of Z on the spin configuration σ
7→ − σ . For any β >
0, the partitionfunction as a function of ( β, J, g ) is defined by Z Λ ( β, J, g ) = X σ ∈ Σ Λ e − βH Λ ( σ,J, g ) , (4)2he average of an arbitrary function f : Σ L → R of the spin configuration in the Gibbs state isgiven by h f ( σ ) i = 1 Z Λ ( β, J, g ) X σ ∈ Σ Λ f ( σ ) e − βH Λ ( σ,J, g ) . Note that the expectation h σ i i of spin at each site i vanishes in the Z symmetric Gibbs state.To study the spontaneous symmetry breaking of the global Z symmetry, assume + boundarycondition, such that σ i = 1 , (5)for i ∈ Λ L \ Λ L − to remove the two-fold degeneracy. Σ +Λ ⊂ Σ Λ denotes a subset of spin configu-rations σ ∈ Σ Λ with this + boundary condition. The phases are classified by the ferromagneticorder parameter m := lim L →∞ | Λ L | X i ∈ Λ L h σ i i , and the Edwards-Anderson spin glass order parameter q := lim L →∞ | Λ L | X i ∈ Λ L h σ i i . Note m ≤ q . The three phases, a Z broken phase m = 0, q = 0, another broken phase m = 0 ,q = 0 and the unique symmetric phase m = q = 0 define the ferromagnetic phase, the spin glassphase and the paramagnetic phase, respectively.To study replica symmetry, define n replicated spin configurations ( σ , · · · , σ n ) ∈ Σ n Λ . Thespin overlap R k,l (1 ≤ k, l ≤ n ) between k -th and l -th spin configurations is defined by R k,l = 1 | Λ L | X i ∈ Λ L σ ki σ li , (6)The bond-overlap is a function of two replicated spin configurations. Here, we consider theHamiltonian as a function of n spin configurations sharing the same random variables g H ( σ , · · · , σ n , J, g ) := n X k =1 H Λ ( σ k , J, g ) (7)Hamiltonian is invariant under any permutation s among n replicated spin configurations. H ( σ s (1) , · · · , σ s ( n ) , J, g ) = H ( σ , · · · , σ n , J, g )This is called replica symmetry. If we calculate the expectation of the site-overlap in the replicasymmetric Gibbs state, it is identical to the Edwards-Anderson spin glass order parameter. h R k,l i = 1 | Λ L | X i ∈ Λ L h σ ki σ li i = 1 | Λ L | X i ∈ Λ L h σ i i = q, The distribution of the site-overlap is broadened in a certain low temperature region includingspin glass phase in the SK model, where the replica symmetric Gibbs state becomes unstable.This phenomenon is RSB, conjectured by Parisi [17] for the SK model, and proven by Talagrand[21]. The condition (5) enables us to detect the finite variance only due to the RSB withoutconfusion due to the Z symmetry. The RSB has been observed in several mean field models[2, 10, 11, 16, 21, 20], while much different properties of the EA model are clarified in thefollowing sections. 3 Ground state
In this section, we review the property of ground state clarified in Ref.[12]
Theorem 3.1
Consider the Edwards-Anderson (EA) model in d -dimensional hyper cubic lattice Λ L under the boundary condition (5). Let f ( σ ) be a real valued function of a spin configuration σ ∈ Σ +Λ . For almost all g , there exists a unique spin configuration σ + ∈ Σ +Λ , such that thefollowing limit is given by lim β →∞ h f ( σ ) i = f ( σ + ) . (8)Theorem implies the following Corollary that RSB does not occur in the EA model in Chat-terjee’s definition [5]. The spin overlap is concentrated at their maximal values in the EA modelat zero temperature for almost all g in any dimensions. Corollary 3.2
In the Edwards-Anderson (EA) model, the following variances of the spin over-lap vanish lim β →∞ E h ( R , − E h R , i ) i = 0 . (9) Proof of Corollary 3.2.
Let us evaluate the expectation value of the site overlap at zerotemperature lim β →∞ h R , i = lim β →∞ | Λ L | X i ∈ Λ L h σ i σ i i = lim β →∞ | Λ L | X i ∈ Λ L h σ i i = 1 | Λ L | X i ∈ Λ L ( σ + i ) = 1 , lim β →∞ h R , i = lim β →∞ | Λ L | X i,j ∈ Λ L h σ i σ i σ j σ j i = lim β →∞ | Λ L | X i,j ∈ Λ L h σ i σ j i = 1 | Λ L | X i,j ∈ Λ L ( σ + i σ + j ) = 1 . (10)Then lim β →∞ h R , i = lim β →∞ h R , i = lim β →∞ h R , i = 1 . (11)These and the dominated convergence theorem implylim β →∞ E h R , i = lim β →∞ E h R , i = lim β →∞ E h R , i = 1 . (12)The variance of the site-overlap vanishes. These complete the proof of Corollary 3.2. (cid:3) roof of Theorem 3.1. First we prove the following identity. Let f ( σ ) be an arbitraryuniformly bounded real valued function of spin configuration σ ∈ Σ Λ , such that | f ( σ ) | ≤ C. Forany bond b ∈ B Λ and for almost all g b , the infinite volume limit and the zero temperature limitof the connected correlation function vanisheslim β →∞ [ h σ b f ( σ ) i − h σ b ih f ( σ ) i ] = 0 . (13)The derivative of one point function gives1 βJ ∂∂g b h f ( σ ) i = h σ b f ( σ ) i − h σ b ih f ( σ ) i . (14)The integration over an arbitrary interval ( g , g ) is1 βJ [ h f ( σ ) i g − h f ( σ ) i g ] = Z g g dg b [ h σ b f ( σ ) i − h σ b ih f ( σ ) i ] . (15)Uniform bounds | f ( σ ) | ≤ C in the left hand side, − C ≤ h σ b f ( σ ) i − h σ b ih f ( σ ) i ≤ C on theintegrand in the right hand side, and the dominated convergence theorem imply the followingcommutativity between the limiting procedure and the integration0 = lim β →∞ Z g g dg b [ h σ b f ( σ ) i − h σ b ih f ( σ ) i ] (16)= Z g g dg b lim β →∞ [ h σ b f ( σ ) i − h σ b ih f ( σ ) i ] . (17)Since the integration interval ( g , g ) is arbitrary, the following limit vanisheslim β →∞ [ h σ b f ( σ ) i − h σ b ih f ( σ ) i ] = 0 , (18)for any b ∈ B Λ for almost all g b ∈ R .Eq.(13) for an arbitrary bond b ∈ B Λ and f ( σ ) = σ b implieslim β →∞ (1 − h σ b i ) = 0 . (19)The above identity can be represented in terms of a probability p b := lim β →∞ h δ σ b , i β →∞ [1 − h (2 δ σ b , − i ] = 1 − (2 p b − = 4 p b (1 − p b ) . (20)Since either p b = 1 or p b = 0 is valid, either a ferromagnetic σ b = 1 or an antiferromagnetic σ b = − b ∈ B Λ for almost all g at zero temperature. The identity (19) indicates the following consistent property between thebond spin configuration and the nearest neighbour spin correlation functions at zero temperature.Consider a plaquette ( i, j, k, l ) with an arbitrary i ∈ Λ L and j = i + e, k = i + e ′ , l = i + e + e ′ for unit vectors with | e | = | e ′ | = 1. Lemma for b = { i, j } , { i, k } and f ( σ ) = σ j σ l , σ k σ l implieslim β →∞ [ h σ i σ j σ j σ l i − h σ i σ j ih σ j σ l i ] = 0 , (21)lim β →∞ [ h σ i σ k σ k σ l i − h σ i σ k ih σ k σ l i ] = 0 . (22)These and σ j = σ k = 1 give the consistent property of the bond spin configurationlim β →∞ h σ i σ j ih σ j σ l ih σ l σ k ih σ k σ i i = 1 . i ∈ Λ L and for b = { i, j } ∈ B Λ , Eq.(13) and f ( σ ) = σ i implylim β →∞ h σ j i = lim β →∞ h σ i σ j ih σ i i = σ { i,j } lim β →∞ h σ i i for almost all g . For any sites i, j ∈ Λ L , i, j are connected by bonds in B Λ . Then, the condition σ i = 1 for i ∈ Λ L \ Λ L − given by the + boundary condition (5) and a bond spin configuration( σ b ) b ∈ B Λ fix a spin configuration σ + ∈ Σ Λ uniquely at zero temperature for any L . This spinconfiguration σ + gives lim β →∞ h f ( σ ) i = f ( σ + ) , for a real valued function f ( σ ) of σ ∈ Σ Λ . This completes the proof of Theorem 3.1. (cid:3) Note that the ferromagnetic order parameter m and the spin glass order parameter q are m = 0 , q = 1 in the spin glass phase and m = 0 , q = 1 in the ferromagnetic phase at zerotemperature. In the previous section, it is shown that the zero temperature infinite volume Gibbs state givesa unique spin configuration in the Edwards-Anderson model with continuous random exchangeinteractions in any dimensions. In this state, the overlap are concentrated at its maximal value.Here, we remark the RSB in mean field spin glass models. For example, the Hamiltonian of theSherrington-Kirkpatrick model defined by H N ( σ ) := − X ≤ i
0, such that P [ R , ∈ ( u − ǫ, u + ǫ ) , H N ( σ ) , H N ( σ ) ≤ ( e + η ) N ] ≥ − Ke − NK , (24)where e := lim N →∞ min σ H N ( σ ) N .
Here, we discuss the energy gap above the unique ground state σ + ∈ Σ +Λ for an arbitraryfixed g in the EA model, where Σ +Λ ( ⊂ Σ Λ ) denotes a set of spin configurations σ satisfyingthe boundary condition (5). It turns out that excited states in the EA model have completelydifferent properties from those in the SK model.For an arbitrary subset S ⊂ Λ ′ L := Λ L \ (Λ L \ Λ L − ), define τ S ∈ Σ +Λ by τ Si = − σ + i for i ∈ S and τ Si = σ + i for i ∈ Λ ′ L \ S . The boundary ∂S of S is a set of bonds defined by ∂S := {{ i, j } ∈ B Λ | i ∈ S, j ∈ Λ ′ L \ S or j ∈ S, i ∈ Λ ′ L \ S } . The energy gap of the spin configuration τ S H Λ ( τ S , J, g ) − H Λ ( σ + , J, g ) = 2 X b ∈ ∂S J g b σ + b , is always positive for any S ⊂ Λ ′ L . 6 heorem 4.1 For any ǫ > and any subset S ( ⊂ Λ ′ L ) ,there exists a function α ( ǫ ) ≥ ǫ andconstants C, e > independent of L, β > , such that E X b ∈ ∂S J g b σ + b = e | ∂S | , (25) E | X b ∈ ∂S J g b σ + b − E X b ∈ ∂S J g b σ + b | ≤ C p | Λ L | /ǫ + | ∂S | α ( ǫ ) , (26)lim ǫ → α ( ǫ ) = 0 . (27)Theorem 4.1, a bound 2 d | S | d − d ≤ | ∂S | ≤ d | S | and Markov’ inequality imply the followingCorollary 4.2 which claims that the energy gap of the spin configuration τ S for R , = 1 isproportional to | ∂S | with probability 1 for d > Corollary 4.2
For r ∈ (0 , , consider a subset S ( ⊂ Λ ′ L ) with | S | = r | Λ L | , which gives theoverlap between σ = σ + and σ = τ S R , = 1 | Λ L | X i ∈ Λ L σ + i τ Si = 1 − | S || Λ L | = 1 − r. For any η > there exists C > independent of L , such that the probability that the energy gapof τ S becomes less than η is bounded by P [ X b ∈ ∂S J g b σ + b < η | ∂S | ] ≤ c ( r ) α ( L − d ) e − η , where c ( r ) := 1 + C (2 d ) d − d ) r − . Proof of Corollary 4.2.
The bound 2 d | S | d − d ≤ | ∂S | ≤ d | S | implies(2 d ) dd − r | Λ L | ≤ | ∂S | dd − , (28)This and the inequality (26) in Theorem 4.1 imply E | X b ∈ ∂S J g b σ + b − E X b ∈ ∂S J g b σ + b | ≤ Cǫ √ r (cid:16) | ∂S | d (cid:17) d d − + | ∂S | α ( ǫ ) . (29)Define ǫ = | ∂S | − d d − . Since ǫ ≤ α ( ǫ ) for sufficiently small ǫ , the bound (26) given by Theorem4.1 implies E | X b ∈ ∂S J g b σ + b − E X b ∈ ∂S J g b σ + b | ≤ | ∂S | [ C (2 d ) d − d ) r − ǫ + α ( ǫ )] (30) ≤ c ( r ) | ∂S | α ( | ∂S | − d d − ) , (31) c ( r ) := C (2 d ) d − d ) r − + 1 . The energy gap (25) in Theorem 4.1 and Markov’s inequality give P [ X b ∈ ∂S J g b σ + b < η | ∂S | ] (32) ≤ P [ | X b ∈ ∂S J g b σ + b − E X b ∈ ∂S J g b σ + b | > E X b ∈ ∂S J g b σ + b − η | ∂S | ] (33) ≤ E | P b ∈ ∂S J g b σ + b − E P b ∈ ∂S J g b σ + b | E P b ∈ ∂S J g b σ + b − η | ∂S | ≤ c ( r ) α ( | ∂S | − d d − ) e − η (34)= c ( r ) α ( L − d ) e − η . (35)7his bound completes the proof of Corollary 4.2. (cid:3) This corollary implies that the finite energy gap of any spin configuration σ = τ S gives theoverlap R , = 1 between the ground state σ = σ + and σ = τ S with probability one for d > σ a ∈ Σ + L , ( a = 1 ,
2) of excited states defined by subsets S a := { i ∈ Λ ′ L | σ ai = σ + i } , with | S a | = r a | Λ L | . The overlap between σ and σ is evaluated as R , = 1 | Λ L | X i ∈ Λ L σ i σ i = 1 | Λ L | [ | Λ L | − | ( S ∪ S ) \ ( S ∩ S ) | ] (36) ≥ − | S | + | S || Λ L | = 1 − r − r . (37)Therefore, Corollary 4.2 implies that any two spin configurations σ , σ with their spin overlap R , < R , = 1 near zero temperatureshould be suppressed by the property of energy gap above the unique ground state in the EAmodel, and thus the absence of RSB in short range spin glass model is predicted also near zerotemperature. To prove Theorem 4.1, here we define several functions in the EA model. The free energy as afunction of ( β, J, g ) is defined byΨ Λ ( β, J, g ) := 1 β log Z Λ ( β, J, g ) . (38)Define p L ( β, J ) := 1 | Λ L | E Ψ Λ ( β, J, g ) , For a subset S ⊂ Λ ′ L , define a subset of bonds B S ⊂ B Λ by B S := {{ i, j } ∈ S | | i − j | = 1 } . (39)Conversely, for a subset A ⊂ B Λ , define a sub-lattice V A ⊂ Λ L by V A := { i ∈ Λ L |∃ b ∈ A ; i ∈ b } (40)Define Hamiltonian by H S ( σ, J, g ) := − X b ∈ B S J g b σ b (41)and define a partition function by Z Λ ,S ( β, J, K, g ) := X σ ∈ Σ S e − β [ H Λ ( σ,J,g )+ H S ( σ,K, g )] , (42)where Σ S := { , − } S . To define its free energyΨ Λ ,S ( β, J, K, g ) = 1 β log Z Λ ,S ( β, J, K, g ) , (43)and P Λ ,S ( β, J, K ) := E Ψ Λ ,S ( β, J, K, g ) . The following Lemma 4.3 and Lemma 4.4 are helpful toprove Theorem 4.1. 8 emma 4.3
The function lim L →∞ p L ( β, J ) converge to p ( β, J ) for each J and uniformly for any β > . Lemma 4.3 is proven by the following inequality. For any
L, M >
0, there exists a constantindependent of
L, M and β , such that he following inequality | p L ( β, J ) − p M ( β, J ) | ≤ C ( J ) (cid:16) L + 1 M (cid:17) . This can be shown in the same way as shown by Chatterjee [5]. Then, the sequence p L ( β, J )becomes Cauchy. Lemma 4.4
Consider a subsystem of the EA model on a sub-lattice S ⊂ Λ ′ L . There exists apositive constant C independent of the system size L and | S | , such that the expectation valueof | Ψ Λ ,S ( β, J, K, g ) − P Λ ,S ( β, J, K ) | has an upper bound E | Ψ Λ ,S ( β, J, K, g ) − P Λ ,S ( β, J, K ) | ≤ C p | Λ L | . Proof of Lemma 4.4.
Denote N := | B Λ | and for an integer m = 1 , , · · · , N , define a symbol E m which denotes the expectation over random variables ( g b ) b>m . Note that E = E is theexpectation over the all random variables g = ( g b ) b =1 , , ··· ,N , and E N is identity.Here, we represent Ψ( g ) = Ψ Λ ,S ( β, J, K, g ) as a function of a sequence of random variables g = ( g b ) b =1 , ··· ,N for lighter notation. E Ψ( g ) − ( E Ψ( g )) (44)= E ( E N Ψ( g )) − E ( E Ψ( g )) (45)= N X b =1 E [( E b Ψ( g )) − ( E b − Ψ( g )) ] . (46)In the b -th term, regard Ψ( g b ) as a function of g b . Let g ′ b be an independent random variablesatisfying the same distribution as that of g b , and E ′ denotes an expectation over only g ′ b . Notethat E b − Ψ( g , · · · , g b , · · · , g N ) = E b E ′ Ψ( g , · · · , g ′ b , · · · , g N ) . Then, each term of the summation (46) for b / ∈ B S becomes E [( E b Ψ( g b )) − ( E b − Ψ( g b )) ] (47)= E [( E b Ψ( g b )) − ( E b E ′ Ψ( g ′ b )) ] = E [ E b (Ψ( g b ) − E ′ Ψ( g ′ b ))] (48)= E [ E b E ′ (Ψ L ( g b ) − Ψ( g ′ b ))] = E h E b E ′ Z g b g ′ b dg ∂∂g Ψ( g ) i , (49)= E h E b E ′ Z g b g ′ b dgJ h σ b i g i ≤ EE b E ′ h Z g b g ′ b dgJ h σ b i g i (50) ≤ EE ′ [ J ( g b − g ′ b )] = 2 J = C b . (51)For b ∈ B S , C b = 2( J + K ) . We denote the Gibbs expectation in the conditional probabilityunder g m by h f ( σ ) i g . Since | B Λ | = | Λ L | d , C := 1 | Λ L | X b ∈ B Λ C b , C , and then E Ψ( g ) − ( E Ψ( g )) ≤ C | Λ L | . (52)This and the Cauchy-Schwarz inequality imply E | Ψ( g ) − E Ψ( g ) | ≤ p E | Ψ( g ) − E Ψ( g ) | E ≤ C p | Λ L | . (53)This completes the proof of Lemma 4.4. (cid:3) Lemma 4.5
Define a real valued function µ : R → R by µ ′ ( g ) = − ρ ( g ) g (54) There exists a real valued function h ( J, K ) , such that the following zero temperature limit isgiven by lim β →∞ E g b h σ b i = 2 µ ( h ( J, K )) . (55) Proof of Lemma 4.5.
Theorem 3.1 implies that h σ b i takes either − h σ b i −∞ = − h σ b i ∞ = 1 and h σ b i is a monotonically increasingfunction of g b . There exists a unique h ∈ R depending on J, K , such that h σ b i has the followinglimt lim β →∞ h σ b i = sgn( g b − h ) . The left hand side is calculated by the above formula.lim β →∞ Z ∞−∞ dg b ρ ( g b ) g b h σ b i = Z ∞−∞ dg b lim β →∞ ρ ( g b ) g b h σ b i (56)= − Z ∞ h dg b µ ′ ( g b ) + Z h −∞ dg b µ ′ ( g b ) = 2 µ ( h ( J, K )) . (57)Since this identity is valid for any other g c for b = c ∈ B L , this is valid also for the expectationover g . This completes the proof of Lemma 4.5. (cid:3) Proof of Theorem 4.1.
Note that for K = 0, e := J E g b h σ b i = 2 µ ( h ( J, b , and then E X b ∈ ∂S J g b h σ b i = | ∂S | e. For a sub-lattice S ⊂ Λ ′ L , define a sub-lattice V := V ∂S ⊂ Λ L . Here, we regard Ψ V ( K ) :=Ψ Λ ,V ( β, J, K, g ) and P V ( K ) := P Λ ,V ( β, J, K, g ) as functions of K >
0. Define a function W ( ǫ )by W ( ǫ ) := 1 ǫ [ | Ψ V ( ǫ ) − P V ( ǫ ) | + | Ψ V (0) − P V (0) | + | Ψ V ( − ǫ ) − P V ( − ǫ ) | ] . (58)Note that Lemma 4.4 gives a bound on the expectation of W ( ǫ ) E W ( ǫ ) ≤ C ǫ p | Λ L | . (59)Convexity of function Ψ V ( K ) and P V ( K ) give | Ψ ′ V (0) − P ′ V (0) | ≤ W ( ǫ ) + P ′ V ( ǫ ) − P ′ V ( − ǫ ) , E | Ψ ′ V (0) − P ′ V (0) | ≤ E W ( ǫ ) + P ′ V ( ǫ ) − P ′ V ( − ǫ ) (60) ≤ C ǫ p Λ L + X b ∈ ∂S ( E g b h σ b i ǫ − E g b h σ b i − ǫ ) (61) ≤ C ǫ p Λ L + 2 X b ∈ ∂S [ µ ( h ( J, ǫ )) − µ ( h ( J, − ǫ ))] (62) ≤ C ǫ p Λ L + α ( ǫ ) | ∂S | , (63)where α ( ǫ ) := 2 µ ( h ( J, ǫ )) − µ ( h ( J, − ǫ )), if 2 µ ( h ( J, ǫ )) − µ ( h ( J, − ǫ )) ≥ ǫ , otherwise α ( ǫ ) := ǫ .Since P V ( K ) is continuously differentiable for almost all K because of its convexity, E g b h σ b i K = µ ( h ( J, K )) is continuous for almost all K . Define C := 3 C , and the limit β → ∞ gives E | X b ∈ ∂S J g b σ + b − E X b ∈ ∂S J g b σ + b | ≤ Cǫ p Λ L + α ( ǫ ) | ∂S | . (64)This completes the proof of Theorem 4.1. (cid:3) References [1] M. Aizenman and P. Contucci, J. Stat. Phys. (3), 641-657 (2010)[4] Arguin, L-P., Damron, M. :On the number of ground states the Edwards-Anderson spinglass model Ann. Inst. H. Poinca´re Probab. Statist. , 93-102 (2015)[6] Edwards,S. F., Anderson, P. W. : Theory of spin glasses J. Phys. F: Metal Phys. , 965-974(1975)[7] Fisher, D. S., Huse, D. A., :Ordered phase of short-range Ising spin glasses Phys. Rev.Lett. , 89-103(1971).[9] S. Ghirlanda and F. Guerra J. Phys. A p -spin models The Ann. of Probab. No3(2014), 946-958.[17] Parisi, G. :A sequence of approximate solutions to the S-K model for spin glasses. J. Phys.A , L115-L121 (1980)[18] Parisi, G., :Order parameter for spin glasses Phys. Rev. Lett. , 1946-1948 (1983).[19] Sherrington, S., Kirkpatrick, S : Solvable model of spin glass. Phys. Rev. Lett. , 1792-1796, (1975).[20] Talagrand, M. : The Parisi formula. Ann. Math.163