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Interaction Flip Identities for non Centered Spin Glasses

We consider spin glass models with non-centered interactions and investigate the effect, on the random free energies, of flipping the interaction in a subregion of the entire volume. A fluctuation bound obtained by martingale methods produces, with the help of integration by parts technique, a family of polynomial identities involving overlaps and magnetizations.


Introduction and main results
The study of factorisation laws for spin glass models has proved to be a fruitful approach to investigate their low temperature phase properties. The introduction of the concept of stochastic stability [AC] and the parallel method of the Ghirlanda-Guerra identities [GG] have in fact received a lot of attention both from the theoretical physics perspective as well as from the purely probabilistic one; in particular it is now well established that stability and factorisation properties are equivalent [PA1,CGG1]. The use of those concepts has led to the final rigorous proof of the Parisi picture for the mean field spin glass, i.e. for the Sherrington-Kirckpatrick model, by the work of Panchenko on ultrametricity [PA2] who heavily relies on factorisation properties.
In the work [CGG2] it was observed that the flip of an interaction in the zero average spin glass case produces, by a suitable use of the integration by parts technique, a set of identities in the deformed state. On the other hand the study of spin glasses is, for physical reasons, interesting also for non-zero values of the interaction average ( [deAT, N]). In this paper we extend the method introduced in [CGG2] to the non zero average case. In order to do so we use the technique of interpolation with the trigonometric method but we also show that the results remain essentially the same also for the linear interpolation method.
We investigate two types of flipping operations. The first flips the entire random interaction while the second only its central part, in both cases in a subregion of the total volume. By comparing the random free energies of a system with given interaction to the one where the interactions have been flipped we show, using a martingale bound on the variance of their difference, that integration by parts produces a family of identities generalizing those found in the zero-mean interaction case. Their first appearance comes from the physical literature where, within the formalism of replica quantum field theory [DeDG, Te], they are refereed as replicon type identities, while the standard stochastic stability argument identifies the longitudinal type identities. As an application, for the spin glass models on a regular lattice in d dimension with periodic boundary conditions, it is interesting to consider a flip of the centered part of the interactions belonging to the boundary hyperplane perpendicular to one of the d directions.
In this case the bound on the variance of the free energies difference implies a bound on the generalized stiffness exponents [FH].
The infinite volume identity obtained by flipping the entire random interaction on a subregion Λ ′ of the lattice Λ reads: provided that in the limit the volume of Λ ′ is non vanishing with respect to that of Λ. In the previous equations m Λ ′ k and c Λ ′ l,k represent normalized magnetization and covariances inside Λ ′ for a set of replicas l, k ... of the system, and the brackets denote the equilibrium states deformed according to the interpolating scheme which determines also the kernel functions h 1 (t, s), h 2 (t, s), k 2 (t, s). Similarly to the case of the Ghirlanda-Guerra identities studied in [CGN], here the effect in the identities derived from spin flip due to the presence of a noncentered disorder is the appearance of terms depending on the system magnetization. The simpler identity (i.e. without magnetization) can be obtained, either by flipping only the centerd part of the disorder, or by taking the integral with respect to a parameter tuning the averages of the disorder. In both cases only the last term of the previous identity survives, the replicon part.
The paper is organised as follows: in the next Section we define the general class of model for which our result apply and we set the notations. In Section 3, after introducing the flipping operations and the corresponding trigonometric interpolations, we give the explicit expressions for the variances of the difference of pressures. In Section 4 the self-averaging theorem is presented. It states that the flip in the sub-volume Λ ′ of the centered part of the disorder produces an effect on the variance which is of the same order of the volume of Λ ′ . On the other hand, the flip of the complete disorder has a non local effect since in this case the variance is bounded by the whole space volume Λ. In the final Section 5 we deduce the identities as consequences of the results of the previous sections. In the Appendix we finally show that making use of a different (e.g. linear) interpolation scheme other identities can be obtained, however the "core" part of the identities involving c Λ ′ 1,2 2 t,s − 2 c Λ ′ 1,2 c Λ ′ 2,3 t,s,t + c Λ ′ 1,2 c Λ ′ 3,4 t,s,s,t is still present.

Definitions
Let introduce the main quantities and the class of models that we will consider.
For every Λ ⊂ Z d let {H Λ (σ)} σ∈Σ N be a family of 2 |Λ| translation invariant (in distribution) Gaussian random variables defined, in analogy with [RU], according to the general representation (σ ∅ = 0) and the J's are independent Gaussian variables with mean Given any subset Λ ′ ⊆ Λ, we also write (2.5) • Average and Covariance matrix.
The Hamiltonian H Λ (σ) has average and covariance matrix (2.7) • Thermodynamic Stability The Hamiltonian is thermodynamically stable if there exists a constantc such that Together with translation invariance a condition like (2.10) is equivalent to where the sum is over the equivalence classeseX of the traslation group. Thanks to the previous relations a thermodynamically stable model fullfills the bound and has an order 1 normalized mean and covariance (2.12) • Average and Covariance matrix in Λ ′ (2.15)

• Parametrized Hamiltonian
In the following we will consider families of random Hamiltonians {X Λ,t (σ)} σ∈Σ N depending on an additional parameter t ∈ I, where I ⊂ R is an interval. For a given inverse temperature β, the corresponding parametrized partition function, pressure, random and quenched measure are defined: where Av (·) is the average with respect to the randomness in X Λ,t .
The measures on the replicated system are defined as usual. For istance, ω t,s := ω t ⊗ ω s is the random interpolated state for a two-copies system and − t,s = Av (ω t,s ) is the corresponding quenched state (the dependence of the quenched states on β will be omitted).
Moreover, we will use the symbol C Λ ′ 1,2 t,s to denote the quenched average of the covariance matrix C Λ ′ (σ, τ ) of two copies of the system labelled by 1 and 2, i.e.: In the same way, we can define the quenched average over three or more copies of the system with respect to any choice of the interpolating parameters. For instance, for an

Spin flip polynomials
In this section we study the effect of flipping the disorder inside a subregion Λ ′ ⊂ Λ. The disorder can be flipped in two ways, the first one being obviuosly: (3.20) The second one can be introduced considering the centered disorder variables {J 0 X } X : and defining the flip as In this second case we restrcit the flip to the centered part of the disorder. The effect of these transformations on the Hamiltonian is obviuos. In fact, denoting with H 0 Λ the hamiltonian with disorder J 0 X , we can write H Λ (σ) = H 0 Λ (σ) + B Λ (σ) and selecting the subvolume Λ ′ , we have Thus the action of the flips on the Hamiltonian are: We are interested in the variation of the random pressure P of H Λ when the Hamiltonian is flipped in both the ways. That is, denoting with P (−) the pressure of we consider In particular, we want to give an explicit expression for the variances of X and X 0 . This can be obtained by introducing an interpolation path, parametrized by t ∈ (a, b), connecting H Λ (σ) Let us now specify the details of the trigonometric interpolations that we will use. Following the approach of [CGG2], we introduce a second family of Gaussian variables J = { J X } X⊂Λ with the same distribution of J and independent of it: and the corresponding hamiltonian where H 0 Λ ′ (σ) is, as in (3.22), the Hamiltonians corresponding to J 0 X := J X − µ X . The interpolation scheme that we will use depends on the flip type that we want to implement.
• Flip F In this case we consider the parametrized Hamiltonian defined, for t ∈ [0, π], by and g(t) := cos t + sin t. Denoting with P(t) the pressure corresponding to X Λ,t (σ), we where ω t (−) is the interpolated random state (2.18) with hamiltonian (3.26).
Lemma 1 For the random variable X 0 we have: and and the interpolated quenched state in (3.31) corresponds to the Hamiltonian (3.28). Proof.
With a simple computation involving the integration by parts for Gaussian random variables we obtain the quenched averages of the hamiltonians: which, substituted in (3.33), give the result (3.30).
The computation of the variances stems from the following formula: The result is obtained by the explicit computation of the averages, which involve a double application of the integration by part formula (3.34). The computation is long but not difficoult; we sketch it for the first term: In the previous line, for the sake of notation, we have introduced the symbol: Applying (3.34) twice (dropping the arguments of B and denoting with δ X,Y the Kronecker symbol), we obtain where the computation of derivatives of B(σ, τ ; t, s) is reduced to that of the Boltzman weights, e.g.: Lemma 2 For the random variable X we have: and where the interpolated quenched state correspond to the Hamiltonian (3.26) and Moreover the kernels in the previous integrals are given by and k 1 (t, s) and k 2 (t, s) are defined in Lemma 1.

Flip of the disorder: self averaging bounds
In this section we will provide bounds for the variance of the random variables X 0 = P − P  Then for every set Λ ′ ⊂ Λ there are positive functions r 0 (β) and r(β) (independent of Λ ′ ?) such that and V (X ) = Av X 2 − Av (X ) 2 ≤ r(β)|Λ|. (4.44)

Proof.
We start by proving the first statement, the second is obtained by a slight modification of the argument. In order to implement the martingale approach devised in [CG1], we enumerate the we can write (we assume that Av 0 means that no integration is performed, while Av N is Av). Therefore the variance of X 0 is Since the sequence {A k } k form a martingale, the sequence Ψ k is a martingale difference, then the Ψ k are mutually orthogonal and where, in the previous notation for the hamiltonians, the dependence over the disorder variables is made explicit and g 0 i (J 0 i ) is the density of the centered Gaussian variable J 0 i , while g ℓ (J ℓ ) is the density of J ℓ . Applying the transformation (3.20) to the J variables in (4.46) and from the symmetry of g 0 i we obtain that (4.46) is transformed into (4.47). Thus, P where V k (−) is the variance with respect to Av k . In order to estimate V k (A k ) for k ≤ M − 1, being obviously zero for k ≥ M , we use an interpolation argument. Thus, we introduce the interpolated hamiltonian on Λ ′ defined as: (4.50) and and ω In order to estimate Av(Ψ 2 k ) we will bound separately the two terms Av[Av k+1 (B ). This remark allows us to integrate with respect to the centered variable J 0 k+1 by appling the integration by parts formula (3.34), which is simpler in the case of the centered variables.

Now we have to estimate Av[Av
Making use of Av k+1 B (the same bound holds for B (−) ).
where Cov k+1 , the covariance w.r.t. J 0 k+1 , can be estimated using the Cauchy-Schwartz inequality. Thus we obtain and recalling (4.49) and (4.52) The sums in the previous inequality are taken over the subsets of Λ ′ , thus we can rewrite them as: Applying the Thermodynamic Stability condition with the formulation (2.9), ane using the which proves the self averaging bounds of X 0 , (4.43) with r 0 (β) = (β 2c + 22 3 β 4c2 ). The proof of (4.44) runs parallel to that of (4.43) up to equation (4.48), which is no loger valid in this case. In fact applying (3.21) the integral (4.46) is not transformed into (4.47), because of the change of sign in B Λ ′ (σ). Thus V (X ) is in general a sum of the N = 2 |Λ| terms Av Ψ 2 k that can be estimated 1 , as in the previous case, by using the thermodynamic stability condition (2.10). As shown above, the result is the existence of a positve function r(β) such that V (X ) ≤ r(β)|Λ|, (4.56) which concludes the proof.
Theorem 1 Suppose that the Hamiltonian (2.1) is thermodynamically stable, see (2.10). Then the following facts hold: 1. Consider the interpolating quenched state corresponding to the Hamiltonian (3.28), then where (c Λ ′ 1,2 ) 2 t,s (and analogously for the other terms) is defined, see (2.15), as while k 2 (t, s) is defined in Lemma 1.

Consider the interpolating quenched state corresponding to the Hamiltonian (3.26), then
for any a > 0 we have where m Λ ′ 1 m Λ ′ 2 t,s (and analogously for the other terms) is defined, see (2.15), as and functions h 1 (t, s), h 2 (t, s), k 2 (t, s) are defined in Lemma 2.

Proof.
Writing (3.31) as a funcion of the normalized quantities (2.11) and (2.12) and using the selfaveraging bound (4.43), we have Then, the first statement follows taking the limit Λ ′ ր Z d .
The proof ot the second statement is similar. We conisder (3.41) with normalized quantities and apply (4.44) obtaining Dividing the two terms of the previous inequality by |Λ| 2 and letting Λ, Λ ′ ր Z d with the constraint |Λ ′ |/|Λ| → a > 0 we obtain the result.
Thus, the complete disorder flip (F) produces an identity which is more complex than the one obtained by flipping the centered part of the disorder only (i.e.F 0 ). Indeed, while (5.57) seems to be independet of the filp type and, to some extent, also on the interpolation path, the identity (5.58) involves extra terms containing the generalised magnetizations m.
It is possible to remove these terms by applying an extra average over the mean value of the disorder variables [CGN] in Λ ′ . In fact, let us introduce a new parameter µ and write the averages of J X as The effect of (5.60) is to introduce a new parameter in the random and interpolating quenched averages. The latter will be denoted by · t;µ or · t,s;µ etc... .
The next result states that, in µ-average, the fluctuations with respect to · t;µ of the generalized magnetization in any macroscopic subregion Λ ′ of Λ is vanishing the large volume limit.
Lemma 3 For every interval [µ 1 , µ 2 ] and any t and Λ ′ ⊂ Λ, we have The proof of this lemma runs parallel, with the obvious modifications, to that of Lemma 4.8 of [CGN] where the fluctuations of the magnetization of the whole region Λ and with respect the quanched state is considered.
It is obvious that Lemma 2, Proposition 4.1 and thus Theorem 1 also hold for the µ-dependent random and quanched measures. Therefore, taking the integral of (3.41) with respect to µ, we obtain the following: Theorem 2 Suppose that the Hamiltonian (2.1) is thermodynamically stable, see (2.10), and consider the interpolating quenched state corresponding to the Hamiltonian (3.26), with the averages of the disorder variables parametrizated by µ, see (5.60). Then for any a > 0 and for any interval [µ 1 , µ 2 ], we have: where k 2 (t, s) is defined in Lemma 2.
Now we want show that the integrals of I 2 (µ), I 3 (µ) vanish in the large volume limit, thus proving (5.62) . In fact, let us consider the covariance Cov t,s;µ (m Λ ′ 1 , m Λ ′ 2 ):= m Λ ′ 1 m Λ ′ 2 t,s;µ − m Λ ′ 1 t;µ m Λ ′ 2 s;µ and write where the Schwarz inequality and (5.61) have been used. In order to bound the integral of I 3 (µ) we introduce the functionm(t, µ) := m Λ ′ t;µ and write where the Schwarz inequality, (5.61) and boundedness of c Λ ′ i,j have been used. Remark. As it was already observed in [CGG2], the terms appearing in the polynomial when expressed in terms of the spin variables, read as: thus ω t,s ((c Λ ′ 1,2 ) 2 ) − 2 ω s,t,s (c Λ ′ 1,2 c Λ ′ 2,3 ) + ω t,s,t,s (c Λ ′ 1,2 c Λ ′ 3,4 ) = (5.64) 1 (with the replica indices dropped). These expressions make clear the correlation-like structure of these quantities. In the case of the Edwards-Anderson model [CG2], (5.64) has a form which is similar to that of spin-glass susceptibility which, in turn, is related to the replicon mass [Te].

Appendix: linear interpolation
In this appendix, to implement the flip F, we consider an interpolation scheme which is simpler than those used in the previous sections since it does not require a second set of disorder variables.
where the interpolated quenched states correspond to the Hamiltonian (6.65).
Proof: same as in Lemma 1.
Arguing as in Theorems 1 and 2, we obtain: