Dynamical Approach to the TAP Equations for the Sherrington-Kirkpatrick Model
Arka Adhikari, Christian Brennecke, Per von Soosten, Horng-Tzer Yau
aa r X i v : . [ m a t h - ph ] F e b Dynamical Approach to the TAP Equations for theSherrington-Kirkpatrick Model
Arka Adhikari ∗ , Christian Brennecke † , Per von Soosten ‡ , Horng-Tzer Yau § Department of Mathematics, Harvard University,One Oxford Street, Cambridge MA 02138, USAFebruary 19, 2021
Abstract
We present a new dynamical proof of the Thouless–Anderson–Palmer (TAP)equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently hightemperature. In our derivation, the TAP equations are a simple consequence ofthe decay of the two point correlation functions. The methods can also be usedto establish the decay of higher order correlation functions. We illustrate this byproving a suitable decay bound on the three point functions from which we derivean analogue of the TAP equations for the two point functions.
We consider systems of N spins σ i , i ∈ { , . . . , N } , taking values in {− , } . TheHamiltonian H N : {− , } N → R of the system is defined by H N ( σ ) = H N ( σ , . . . , σ N ) = X ≤ i 1) is a standard Gaussianrandom variable. Physically, the value q ∈ [0; 1] corresponds to the limiting value ofthe overlap distribution in the replica-symmetric high temperature regime. The overlap R , : {− , } N × {− , } N → R is defined by R , ( σ , σ ) = 1 N N X i =1 σ i σ i . Its distribution under E h·i ⊗ h·i is the functional order parameter of the system in thethermodynamic limit N → ∞ . At sufficiently high temperature, the overlap distributionis expected to concentrate on a single point. In fact, for t < / 4, one can prove thatthe overlap concentrates exponentially, which is a key input for a detailed mathematicalunderstanding of the Gibbs measure at high temperature (see [22, Sections 1.4 to 1.11]).The validity of the TAP equations (1.2) at high temperature has been establishedby Talagrand [20, 22] and Chatterjee [6]. Both works rely on the concentration of theoverlap as a key ingredient in the proof. More recently, Bolthausen [4, 5] constructedan iterative solution of the TAP equations in the full high temperature regime andused it to provide a new proof of the replica-symmetric formula for the free energy atsufficiently high temperature. In [7], Chen and Tang proved that Bolthausen’s schemeindeed approximates the magnetizations of the SK model, assuming locally uniformconcentration of the overlap. At low temperature, Auffinger and Jagannath [3] proved aversion of the TAP equations for generic mixed p -spin models. In this case, the overlapis not a constant anymore, but one can decompose the hypercube into clusters (“pure2tates”) within which the overlap remains approximately constant. Then, the TAPequations remain valid conditionally on each cluster (see [3] for more precise details).An interesting open problem is to prove the replica-symmetry of the SK model inthe full high temperature regime predicted by de Almeida and Thouless [1]. The systemis believed to be replica-symmetric for all ( t, h ) that satisfy E t cosh ( √ tqZ + h ) < , (1.3)where q = E tanh ( √ tqZ + h ) and Z ∼ N (0 , 1) as above. In particular, the TAPequations (1.2) are believed to be valid under the AT condition (1.3). So far, replica-symmetry is known above the AT line up to a bounded region in the ( t, h )-phase diagram.This has been proved in [11] through an analysis of the Parisi variational problem.The goal of this work is to present a new proof of the TAP equations that relies ona direct dynamical approach by viewing the couplings g ij as Brownian motions runningat speed 1 /N . After applying Itˆo’s lemma to the magnetizations, this point of viewleads naturally to a dynamical study of the two point functions m ij . For sufficientlyhigh temperature, we prove suitable decay bounds on the m ij from which the TAPequations follow with explicit error bounds as a simple corollary. Our approach extendsto higher order correlation functions in a straightforward way. In particular, we provean analogue of the TAP equations for the two point functions which provides a simpleheuristic connection to the AT condition (1.3). For this reason, we hope that a dynamicalapproach will contribute to an improved understanding of the high temperature regime.Tools from stochastic calculus have provided useful insights into the probabilisticstructure of the SK model in the past. Comets and Neveu [8] gave an elegant newproof of the fundamental high temperature results of Aizenman, Lebowitz and Ruelle[2] in the absence of an external field by representing the partition function as a suitablestochastic exponential and invoking a martingale central limit theorem. Moreover, theinterpolation method of Guerra, whose core mechanism is based on Gaussian integrationby parts, can also be rewritten dynamically in terms of Itˆo’s lemma. The paper of Tindel[25] combines the previous two perspectives to extend the central limit theorem for thefree energy to a region with positive external field strength. In contrast to these works,our present approach directly tracks the evolution of the magnetization and higher ordercorrelation functions as the coupling strengths between one particle and the others aregradually increased. This approach gives rise to the TAP equations in a natural fashionand makes the corresponding computations for the higher order correlation functionssystematic and tractable.For the statement of our main results, let m ( i ) k and m ( i ) kl denote the magnetizationsand two point correlation functions, respectively, after the i -th particle σ i has beenremoved from the N -spin system (see the next section for a precise definition). Ourmain result describes the validity of a hierarchical version of the TAP equations (alsocalled the cavity equations) for all 0 ≤ t < log 2 in the sense of L ( P ).3 heorem 1.1. Let ≤ t < log 2 . Then, there exists a constant C = C t > , independentof N ∈ N , such that E (cid:20) m i − tanh (cid:16) h + X j = i g ij m ( i ) j (cid:17)(cid:21) ≤ CN . (1.4) Moreover, for all ǫ > sufficiently small and i = j , there exists C = C t,ǫ > such that E (cid:20) m ij − (cid:18) − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:19) X l = i g il m ( i ) lj (cid:21) ≤ CN ǫ . (1.5)We point out that equation (1.4) for the magnetizations has been studied beforein [22, Lemma 1.7.4], where a similar bound is proved for t < / 4. In fact, (1.4) is whatone would expect from the classical heuristic m i = D sinh (cid:16) h + P j = i g ij σ j (cid:17) E ( i ) D cosh (cid:16) h + P j = i g ij σ j (cid:17) E ( i ) ≈ tanh (cid:18) h + X j = i g ij m ( i ) j (cid:19) for a mean-field ferromagnet, which is correct (at least) when the spins are approximatelyindependent under the Gibbs measure. However, unlike the ferromagnetic case, thetypical size of the couplings g ij = O ( N − / ) and the correlations between g ij and m ( i ) j prohibit one from obtaining the classical mean–field equations by inserting the heuristic m ( i ) j ≈ m j . Instead, this substitution results in the Onsager correction t (1 − q ) m i in theTAP equations. The significance of (1.4) and (1.5) is that they display the leading orderdependence of m i and m ij on the i -th column ( g ik ) ≤ k ≤ N of the interaction. Notice that,on a heuristic level, the equations (1.5) for the m ij follow simply by differentiation of theTAP equations (1.4) for the m i with respect to the external field. Alternatively, (1.5)can also be derived using a cavity field heuristic, see [12, Section V.3].As already observed in [12, Section V.3], it is interesting to note that the hierarchicalTAP equations for the one and two point functions have a simple connection to the ATcondition (1.3). To see this, let us assume that q N = 1 N N X k =1 m k ≈ N N X k =1 (cid:0) m ( i ) k (cid:1) = q ( i ) N , which follows from the decay of correlations and let us assume in addition that q N = 1 N N X k =1 m k ≈ E N N X k =1 m k . (1.6)Notice that this concentration assumption is reasonable since q N = 1 N N X k =1 m k = h R , i . 4e then conclude from the TAP equations (1.4) and (1.6) that q N ≈ E tanh (cid:16) h + X j = i g ij m ( i ) j (cid:17) = E tanh (cid:16) h + q tq ( i ) N Z i (cid:17) ≈ E tanh ( h + √ tq N Z i )for the standard Gaussian Z i = (cid:0) tq ( i ) N (cid:1) − / P k = i g ik m ( i ) k ∼ N (0 , q N ≈ q is close to the unique fixed point q = E tanh ( h + √ tqZ ). Based on Theorem1.1, we will make this rigorous and prove the following concentration result. Proposition 1.2. Let ≤ t < log 2 and let q = E tanh ( h + √ tqZ ) , where Z ∼ N (0 , denotes a standard Gaussian random variable. Let q N = N − P Nk =1 m k , then there existsa constant C = C t > such that E | q − q N | ≤ CN / . (1.7)If we use the information of Proposition 1.2 and assume in addition that mixedmoments of distinct correlation functions are of lower order o ( N − ), we recover theAT transition line (1.3) as a singularity in the norm of the two point functions. Moreprecisely, applying (1.5), we obtain from Gaussian integration by parts and separatingthe diagonal term in the sum P l = i (cid:0) m ( i ) lj (cid:1) that E m ij ≈ E t (cid:18) − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:19) N X l = i (cid:0) m ( i ) lj (cid:1) + E t N X l ,l = i (cid:20) ∂ il ∂ il (cid:18) − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:19) (cid:21) m ( i ) l j m ( i ) l j ≈ E t (cid:2) − tanh ( h + √ tqZ (cid:1)(cid:3) (cid:20) N E (cid:16) − (cid:0) m ( i ) j (cid:1) (cid:17) + E N X l = i,j (cid:0) m ( i ) lj (cid:1) (cid:21) + o ( N − ) ≈ tN (cid:20) E ( h + √ tqZ (cid:1) (cid:21) + E t cosh ( h + √ tqZ (cid:1) E m ij . (1.8)Here, we used the approximation E (cid:0) m ( i ) lj (cid:1) ≈ E m lj , which will be justified later. More-over, we used that Z i = (cid:0) tq ( i ) N (cid:1) − / X k = i g ik m ( i ) k is independent of the remaining disorder g kl , for k, l = i , because of the Gaussian struc-ture (see also [22, Lemma 1.7.6]). Altogether, we expect thatlim N →∞ E (cid:0) √ N m ij (cid:1) = t (cid:20) − E t cosh ( h + √ tqZ (cid:1) (cid:21) − (cid:20) E ( h + √ tqZ (cid:1) (cid:21) , where the right hand side is finite if (1.3) holds true. Based on (1.8) as well as the resultsof Theorem 1.1 and Proposition 1.2, we will prove the following proposition.5 roposition 1.3. Let ≤ t < log 2 and let q = E tanh ( h + √ tqZ ) , where Z ∼ N (0 , denotes a standard Gaussian random variable. Then, for every ǫ > sufficiently small,there exists a constant C = C ε,t > so that E m ij = tN (cid:20) − E t cosh ( h + √ tqZ (cid:1) (cid:21) − (cid:20) E ( h + √ tqZ (cid:1) (cid:21) + Θ (1.9) for an error Θ bounded by | Θ | ≤ C/N ǫ . The leading order behavior (1.9) of the two point functions m ij is well-known andalready mentioned in [12, Section V.3]. A rigorous proof of the identity (1.9) for t < / m ij were analyzed in [10].These proofs are, however, not based on the heuristics outlined in (1.8).As a corollary of Theorem 1.1, we are also able to derive the TAP equations. Corollary 1.4. Let ≤ t < log 2 . Then, there exists a constant C = C t > , indepen-dent of N ∈ N , such that E (cid:20) m i − tanh (cid:16) h + X j = i g ij m j − t (1 − q N ) m i (cid:17)(cid:21) ≤ CN , (1.10) where q N is defined by q N = N − P Nk =1 m k .Moreover, for any ǫ > sufficiently small, there exists C = C t,ǫ > such that E (cid:20) m ij − (cid:0) − m i (cid:1)(cid:18) X k = i g ik m kj + 2 tN ( M m ) j m i − t (1 − q N ) m ij (cid:19)(cid:21) ≤ CN ǫ (1.11) for all i = j . Here, we set M = ( m kl ) ≤ k,l ≤ N and m = ( m , . . . , m N ) . We point out that, using Proposition 1.2, we can replace q N in (1.10) by the solution q = E tanh ( h + √ tqZ ), up to another error that vanishes as N → ∞ . This yields (1.10)in the form that is typical in the mathematical literature on the subject. Remark 1.5. Let us mention that (1.11) represents a resolvent equation for the matrix M = ( m kl ) ≤ k,l ≤ N . Indeed, neglecting the error terms, (1.11) means that M ≈ − tA − G − E , (1.12) where Λ ij = (1 − m i ) − δ ij , A ij = 2 N − m i m j , E = − t (1 − q N ) ≈ − t (1 − q ) and G consists of the couplings { g ij } extended to a symmetric matrix. Thus, one recoversthe resolvent of a deformed Gaussian Orthogonal Ensemble at the energy E . Like theheuristics following Theorem 1.1, this suggests to study the high temperature regime inview of the singularity of M , a viewpoint reminiscent of [18] (see also [18, Eq. (3.3)]).Based on the observation in (1.12) , the AT condition can also be expressed in termsof a spectral condition. To see this, let us neglect the rank-one perturbation A and he correlations between Λ and G , which should be weak at high temperature. Setting G = √ t e G for a GOE matrix e G , we are evaluating M ( E ) = (cid:16) Λ − √ t e G − E (cid:17) − at a special energy E = − t (1 − q ) . From random matrix theory we expect M ii ( E ) = 1Λ ii − E − tS ( E ) with S ( E ) = 1 N X i ii − E − tS ( E ) . Here, E can be real as long as it is outside of the spectrum. Now notice that S ′ ( E ) = (1 + tS ′ ( E )) 1 N X i ii − E − tS ( E )) = (1 + tS ′ ( E )) 1 N X i ( M ii ( E )) . If we plug in E = − t (1 − q ) , this calculation says that S ′ ( E ) = (1 + tS ′ ( E )) 1 N N X i =1 (1 − m i ) ≈ (1 + tS ′ ( E )) E sech ( h + √ tqZ ) , so that S ′ ( E ) = E sech ( h + √ tqz )1 − t E sech ( h + √ tqz ) . In particular, S ′ ( E ) is finite precisely under the AT condition. Since S is supposed tobe analytic everywhere except the spectral edge, this fits in nicely with E being outsidethe spectrum under the AT condition. Let us conclude this introduction with some comments about how to extend ourresults to mixed p -spin models. To this end, let H ( p ) N : {− , } N → R be defined by H ( p ) N ( σ ) = h + β ∞ X p =2 β p √ p ! N ( p − / X | A | = p g A Y i ∈ A σ i for i.i.d. standard Gaussian random variables ( g A ) A ⊂{ ,...,N } and a sequence ( β p ) p ≥ ensuring that ξ ( s ) := β P ∞ p =2 β p s p < ∞ for all s ∈ [0; 1]. The function ξ characterizesthe model in the sense that E ( H ( p ) N ( σ ) − h )( H ( p ) N ( σ ) − h ) = ξ (cid:0) R , ( σ , σ ) (cid:1) . β ≥ β p = β p for some β ≥ C = C β,β > E (cid:20) m i − tanh (cid:18) h + β ∞ X p =2 β p √ p ! N ( p − / X i ∈ A, | A | = p g A D Y k ∈ A,k = i σ k E ( i ) (cid:19)(cid:21) ≤ CN , E (cid:20) m ij − sech (cid:18) h + β ∞ X p =2 β p √ p ! N ( p − / X i ∈ A, | A | = p g A D Y k ∈ A,k = i σ k E ( i ) (cid:19) × β ∞ X p =2 β p √ p ! N ( p − / X i ∈ A, | A | = p g A D σ j ; Y k ∈ A,k = i σ k E ( i ) (cid:21) ≤ CN ǫ (1.13)for all ǫ > i = j . Here, we denote D σ j ; Y k ∈ A,k = i σ k E ( i ) = D σ j Y k ∈ A,k = i σ k E ( i ) − h σ j i ( i ) D Y k ∈ A,k = i σ k E ( i ) and all Gibbs expectations are taken with respect to the Gibbs measure induced by H ( p ) N . Since the methods to prove Theorem 1.1 can be adapted in a straight-forward wayto prove the bounds in (1.13), we focus in this paper exclusively on the analysis of the2-spin model with Hamiltonian H N defined in (1.1).Finally, let us remark that also the heuristics in (1.8) can be generalized to the p -spinmodels. Indeed, let us assume appropriate decay of correlations so that we can factorize D Y k ∈ A,k = i σ k E ( i ) ≈ Y k ∈ A,k = i m ( i ) k . Writing A = { j , j , . . . , j p } , this can be made rigorous by using the identity D Y k ∈ A,k = i σ k E ( i ) − Y k ∈ A,k = i m ( i ) k = D σ j ; Y k ∈ A,k = i,j σ k E ( i ) + D σ j ; Y k ∈ A,k = i,j ,j σ k E ( i ) m ( i ) j + . . . + m ( i ) j p − j p Y k ∈ A,k = i,j p − ,j p m ( i ) k and adapting the methods presented below to show that the correlation functions on theright hand side are small in the limit N → ∞ . By (1.13) and in analogy to Prop. 1.2,we then expect that q N = N − P Nk =1 m k ≈ E q N concentrates and converges as N → ∞ to a solution q ∈ [0; 1] of the self-consistent equation q = E tanh ( h + p ξ ′ ( q ) Z ) . Here, Z ∼ N (0 , 1) denotes a standard Gaussian. Assuming similarly that D σ j ; Y k ∈ A,k = i σ k E ( i ) ≈ X k ∈ A,k = i m jk Y l ∈ A,l = i,k m l , 8e may follow the heuristics of (1.8) and expect that E m ij ≈ N E sech ( h + p ξ ′ ( q ) Z (cid:1) E ξ ′′ ( q ) sech ( h + p ξ ′ ( q ) Z (cid:1) − E ξ ′′ ( q ) sech ( h + p ξ ′ ( q ) Z (cid:1) . In particular, this can only hold true under the condition E ξ ′′ ( q ) sech ( h + p ξ ′ ( q ) Z (cid:1) < , which appears to be consistent with the generalized AT condition that is conjecturedin [11, Eq. (1.8) & Eq. (1.9)] (assuming that, at sufficiently high temperature, theself-consistent equation q = E tanh ( h + p ξ ′ ( q ) Z ) has a unique fixed point).The paper is structured as follows. In the following Section 2, we introduce ournotation. In Section 3, we establish suitable decay bounds on the two and three pointcorrelation functions. In Sections 4 and 5, we prove the TAP equations in the sense ofTheorem 1.1 and Corollary 1.4. Finally, in Section 6, we prove Propositions 1.2 and 1.3. Acknowledgements. The work of P. S. is supported by the DFG grant SO 1724/1-1. Part of this work was written when A.A. was under the sponsorship of a HarvardUniversity GSAS MGSTTRF. The research of H.-T. Y. is partially supported by NSFgrant DMS-1855509 and a Simons Investigator award. In the following, we will need to consider expectations of observables conditionally ona given number of spins. To this end, it is useful to set up the following notation. Let A = { j , j , . . . , j k } ⊂ { , . . . , N } , let B ⊂ { , . . . , N } be disjoint from A with | B | = l and let τ = ( τ j , . . . , τ j k ) ∈ {− , } k be a fixed j -particle configuration. Then, we definethe reduced Hamiltonian H [ A,B ] N ≡ H [ A,B ] N, ( τ j ,...,τ jk ) : {− , } N − k − l → R by H [ A,B ] N ( σ ) = H [ A,B ] N ( σ i , . . . , σ i N − k − l ) = X ≤ i 1) By optimizing the Gronwall argument from the previous proof, one can improvethe lemma to hold for all times t ≥ ≤ t < max x ∈ [2; ∞ ) x log (cid:20) x − (cid:16) 13 + x (cid:17) − (cid:16) 23 + 23 x (cid:17) − (cid:21) ≈ . . 2) The bound provided in Lemma 3.1 is clearly uniform in time. More precisely, wehave sup s ij ∈ [0; t ] , ≤ i Let ≤ t < log 2 , let A ⊂ { , . . . , N } and choose ǫ > sufficiently small.Then, for some C ǫ > , independent of N , t and A ⊂ { , . . . , N } , we have that sup σ ∈{− , } | A | E (cid:12)(cid:12) m [ A ] ijk (cid:12)(cid:12) ≤ C t,ǫ N ǫ/ for all i = j, i = k, j = k and i, j, k A .Proof. Lemma (3.1) and the Cauchy-Schwarz inequality combine to show that (cid:13)(cid:13) m [ A ] ik δ i m [ A ∪{ i } ] j (cid:13)(cid:13) ≤ (cid:13)(cid:13) m [ A ] ik (cid:13)(cid:13) (cid:13)(cid:13) δ i m [ A ∪{ i } ] j (cid:13)(cid:13) ≤ (cid:13)(cid:13) m [ A ] ik (cid:13)(cid:13) ǫ/ ǫ (cid:13)(cid:13) δ i m [ A ∪{ i } ] j (cid:13)(cid:13) ǫ/ ǫ ≤ C t,ǫ N ǫ/ . (3.5)By the identity (3.2), it is therefore enough to control δ i m [ A ∪{ i } ] jk . Differentiating theidentity (3.3) with respect to the external field in the direction of σ k , we find that δ i m [ A ∪{ i } ] jk = X l A Z t ε i m [ A ∪{ i } ] jkl ( s ) dg il ( s ) − X l A Z t δ i (cid:16) m [ A ∪{ i } ] kl m [ A ∪{ i } ] jl (cid:17) ( s ) dsN − X l A Z t δ i (cid:16) m [ A ∪{ i } ] l m [ A ∪{ i } ] jkl (cid:17) ( s ) dsN . We proceed as in Lemma 3.1, using the Itˆo isometry followed by Young’s inequality. Ifwe also apply the trivial bound to the summands with l ∈ { j, k } and insert the bounds13f Lemma 3.1 for the two-point functions, we conclude that E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ≤ C t,ǫ N ǫ/ + X l A ∪{ j,k } Z t E (cid:12)(cid:12) ε i m [ A ∪{ i } ] jkl (cid:12)(cid:12) ( s ) dsN − X l A ∪{ j,k } Z t E (cid:0) δ i m [ A ∪{ i } ] jk (cid:1) ( s ) δ i (cid:16) m [ A ∪{ i } ] l m [ A ∪{ i } ] jkl (cid:17) ( s ) dsN ≤ C t,ǫ N ǫ/ + Z t E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ( s ) ds + sup l A ∪{ j,k } ,σ i = ± Z t E (cid:16) (cid:12)(cid:12) m [ A ∪{ i } ] l (cid:12)(cid:12) (cid:17)(cid:12)(cid:12) m [ A ∪{ i } ] jkl (cid:12)(cid:12) ( s ) ds, uniformly in A ⊂ { , . . . , N } . Using once more the identity (3.2) together with theresults of Lemma 3.1 and the remarks following its proof, we have thatsup s ∈ [0; t ] E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ A ∪{ i } ] jkl (cid:12)(cid:12) ( s ) − h − (cid:12)(cid:12) m [ A ∪{ i } ] l (cid:12)(cid:12) i (cid:12)(cid:12) δ l m [ A ∪{ i,l } ] jk (cid:12)(cid:12) ( s ) (cid:12)(cid:12)(cid:12) ≤ C t,ǫ N ǫ/ and hence E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ≤ C t,ǫ N ǫ/ + Z t E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ( s ) ds + sup l A ∪{ j,k } ,σ i = ± Z t E (cid:12)(cid:12) δ l m [ A ∪{ i,l } ] jk (cid:12)(cid:12) ( s ) ds. Gronwall’s Lemma implies that E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ≤ C t,ǫ N ǫ/ + sup l A ∪{ j,k } ,σ i = ± Z t e t − s E (cid:12)(cid:12) δ l m [ A ∪{ i,l } ] jk (cid:12)(cid:12) ( s ) ds and by iterating this estimate N − | A | times, as in the proof of Lemma 3.1, we find E (cid:12)(cid:12) δ i m [ A ∪{ i } ] jk (cid:12)(cid:12) ≤ C t,ǫ N ǫ/ for t < log 2, uniformly in A ⊂ { , . . . , N } . Together with (3.5), this proves the claim. Remark: 1) Viewing the ( g ij ) ≤ i Using the bounds on the size of the correlation functions, we are now ready to provethe validity of the hierarchical TAP equations for the one and two point functions in thesense of L ( P ). This will prove in particular our main result Theorem 1.1. Lemma 4.1. Let ≤ t < log 2 . Then, for some C = C t > independent of N , we have E h m i − tanh (cid:16) h + X j = i g ij m ( i ) j (cid:17)i ≤ CN . Proof. By the Lipschitz continuity of tanh( · ), the claim follows if we show that E h tanh − ( m i ) − (cid:16) h + X j = i g ij m ( i ) j (cid:17)i ≤ CN . We view the ( g ij ) ≤ j ≤ N dynamically as Brownian motions at time t and of speed 1 /N so that a straight forward application of Itˆo’s Lemma implies thattanh − ( m i ) − (cid:16) h + X j = i g ij m ( i ) j (cid:17) = X j = i Z t (cid:18) m j − m ( i ) j − m i m ij − m i (cid:19) ( s ) dg ij ( s ) − Z t X j = i (cid:18) m j m ij − m i − m i m ij − m i − m i m ij (1 − m i ) (cid:19) ( s ) dsN . Recalling that m ij / (1 − m i ) = δ i m [ i ] j , we use | m i | ≤ | δ i m [ i ] j | ≤ (cid:13)(cid:13)(cid:13)(cid:13) Z t X j = i (cid:18) m j m ij − m i − m i m ij − m i − m i m ij (1 − m i ) (cid:19) ( s ) dsN (cid:13)(cid:13)(cid:13)(cid:13) ≤ C X j = i Z t k δ i m [ i ] j ( s ) k dsN By the observation (3.4) after the proof of Lemma 3.1, this implies that E (cid:20) Z t X j = i (cid:18) m j m ij − m i − m i m ij − m i − m i m ij (1 − m i ) (cid:19) ( s ) dsN (cid:21) ≤ CN . (4.1)Similarly, it follows that E (cid:20) Z t X j = i m i m ij − m i ( s ) dg ij ( s ) (cid:21) ≤ C X j = i Z t E (cid:12)(cid:12) δ i m [ i ] j ( s ) (cid:12)(cid:12) dsN ≤ CN . (4.2)Hence, it remains to control the size of E (cid:20) Z t X j = i (cid:0) m j − m ( i ) j (cid:1) ( s ) dg ij ( s ) (cid:21) = Z t X j = i E (cid:0) m j − m ( i ) j (cid:1) ( s ) dsN . 15o this end, we use that m j = h σ j i = (cid:10) m [ i ] j (cid:11) so that E (cid:18) Z t X j = i (cid:0) m j − m ( i ) j (cid:1) ( s ) dg ij ( s ) (cid:19) ≤ tN X j = i sup s ∈ [0; t ] sup σ i = ± E (cid:0) m [ i ] j − m ( i ) j (cid:1) ( s ) . Applying once more Itˆo’s Lemma yields (cid:0) m [ i ] j − m ( i ) j (cid:1) ( s ) = σ i Z s X k = i m [ i ] jk ( u ) dg ik ( u ) − Z s X k = i m [ { i } ] k m [ { i } ] jk ( u ) duN , so that the estimate (3.4) impliessup s ∈ [0; t ] sup σ i = ± E (cid:0) m [ i ] j − m ( i ) j (cid:1) ( s ) ≤ CN . Thus, we find that E (cid:18) Z t X j = i (cid:0) m j − m ( i ) j (cid:1) ( s ) dg ij ( s ) (cid:19) ≤ CN and together with the bounds (4.1), (4.2), this proves the claim.In order to prove the analogue of the hierarchical TAP equations for the two pointfunctions, we also need the bounds from Lemma 3.2 and the remark following its proof. Lemma 4.2. Let ≤ t < log 2 and assume ǫ > to be sufficiently small. Then, forsome C = C t,ǫ > , independent of N , we have that E (cid:20) m ij − (1 − m i ) X k = i g ik m ( i ) kj (cid:21) ≤ CN ǫ/ . Proof. We consider the ( g ik ) ≤ k ≤ N dynamically and use Itˆo’s Lemma to compute m [ i ] j ( t ) = m ( i ) j + X k = i Z t σ i m [ i ] kj ( s ) dg ik ( s ) − X k = i Z t m [ i ] k m [ i ] kj ( s ) dsN = m ( i ) j + X k = i σ i g ik m ( i ) kj − X k = i m ( i ) k m ( i ) kj tN + X k = i Z t σ i (cid:0) m [ i ] kj − m ( i ) kj (cid:1) ( s ) dg ik ( s ) − X k = i Z t (cid:0) m [ i ] k m [ i ] kj ( s ) − m ( i ) k m ( i ) kj (cid:1) dsN . (4.3)16f we average the last equation over the spin variable σ i ∈ {− , } and multiply itafterwards by (1 − m i ), we find with the identity (3.1) that (cid:13)(cid:13)(cid:13) m ij − (1 − m i ) X k = i g ik m ( i ) kj (cid:13)(cid:13)(cid:13) ≤ sup σ i = ± (cid:13)(cid:13)(cid:13) X k = i Z t (cid:0) m [ i ] kj − m ( i ) kj (cid:1) ( s ) dg ik ( s ) (cid:13)(cid:13)(cid:13) + sup σ i = ± (cid:13)(cid:13)(cid:13) X k = i Z t (cid:0) m [ i ] k m [ i ] kj ( s ) − m ( i ) k m ( i ) kj (cid:1) dsN (cid:13)(cid:13)(cid:13) ≤ CN / sup s ∈ [0; t ] sup σ i = ± (cid:13)(cid:13) m [ i ] j ( s ) − m ( i ) j (cid:13)(cid:13) + sup s ∈ [0; t ] sup k = i,jσ i = ± (cid:13)(cid:13) m [ i ] kj ( s ) − m ( i ) kj (cid:13)(cid:13) + sup s ∈ [0; t ] sup k = i,jσ i = ± (cid:13)(cid:13)(cid:0) m [ i ] k m [ i ] kj (cid:1) ( s ) − m ( i ) k m ( i ) kj (cid:13)(cid:13)(cid:13) . (4.4)Hence, let us bound the norms on the right hand side to conclude the claim.First of all, a straight forward application of Lemma 3.1 and Eq. (4.3) implies that CN / sup s ∈ [0; t ] sup σ i = ± (cid:13)(cid:13) m [ i ] j ( s ) − m ( i ) j (cid:13)(cid:13) ≤ CN . For the two other error terms, we use again Itˆo’s Lemma which shows that m [ i ] jk ( s ) − m ( i ) jk = X l = i Z s σ i m [ i ] jkl ( u ) dg il ( u ) − X l = i Z s (cid:0) m [ i ] l m [ i ] jkl + m [ i ] jl m [ i ] kl (cid:1) ( u ) duN (4.5)and, by the product rule, that (cid:0) m [ i ] k m [ i ] jk (cid:1) ( s ) − m ( i ) k m ( i ) jk = X l = i Z s σ i m [ i ] jk ( u ) m [ i ] kl ( u ) dg il ( u ) − X l = i Z s (cid:0) m [ i ] l m [ i ] jk m [ i ] kl (cid:1) ( u ) duN + X l = i Z s σ i m [ i ] k ( u ) m [ i ] jkl ( u ) dg il ( u ) − X l = i Z s m [ i ] k ( u ) (cid:0) m [ i ] l m [ i ] jkl + m [ i ] jl m [ i ] kl (cid:1) ( u ) duN + X l = i Z s m [ i ] kl ( u ) m [ i ] jkl ( u ) duN . (4.6)17sing Lemmas 3.1, 3.2 and the remarks following their proofs, it is simple to check thatsup s ∈ [0; t ] sup k = i,jσ i = ± (cid:13)(cid:13) m [ i ] kj ( s ) − m ( i ) kj (cid:13)(cid:13) + sup s ∈ [0; t ] sup k = i,jσ i = ± (cid:13)(cid:13)(cid:0) m [ i ] k m [ i ] kj ( s ) − m ( i ) k m ( i ) kj (cid:1) ( s ) (cid:13)(cid:13)(cid:13) ≤ C ǫ N / ǫ/ . Plugging these estimates into (4.4), we conclude the lemma.We conclude this section with the proof of Theorem 1.1. Proof of Theorem 1.1. Lemma 4.1 establishes the hierarchical TAP equations (1.4), soit only remains to prove the bound (1.5). This is a simple consequence of Lemmas 3.1,4.1 and 4.2. Indeed, using Cauchy-Schwarz we find that E (cid:20) m ij − (cid:18) − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:19) X l = i g il m ( i ) lj (cid:21) ≤ C E (cid:20)(cid:18) m i − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:19) X l = i g il m ( i ) lj (cid:21) + CN ǫ/ ≤ C (cid:13)(cid:13)(cid:13) m i − tanh (cid:16) h + X k = i g ik m ( i ) k (cid:17)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) X k = i g ik m ( i ) kj (cid:13)(cid:13)(cid:13) + CN ǫ/ ≤ CN ǫ/ . In this section, we prove the bounds (1.10) and (1.11) from Corollary 1.4. Proof of Corollary 1.4. We begin with the proof of (1.10). We have to compute theleading order contribution to X k = i g ik (cid:0) m k − m ( i ) k (cid:1) =: X k = i g ik W k . To compute the leading order, we view W k = W k ( g ik ) as a function of the coupling g ik and we do a second order Taylor expansion. This implies that W k ( g ik ) = W k ( g ik = 0) + g ik m i (1 − m k )( g ik = 0) − g ik ( m k m ik )( g ik = 0)+ g ik Z ds Z s ds (cid:0) ∂ ik m k (cid:1) ( s g ik ) . X k := W k − m i (1 − m k ) g ik , we thus obtain that X k = W k ( g ik = 0) − g ik ( m k m ik )( g ik = 0) − g ik Z ds (cid:0) ∂ ik ( m i (1 − m k )) (cid:1) ( sg ik )+ g ik Z ds Z s ds (cid:0) ∂ ik m k (cid:1) ( s g ik ) . Next, let us prove that E (cid:18) X k = i g ik X k (cid:19) ≤ CN . (5.1)Since | ∂ ik ( m i (1 − m k ))( sg ik ) | ≤ C, | ∂ ik m k ( sg ik ) | ≤ C (uniformly in s ∈ [0; 1]), we have E (cid:18) X k = i g ik Z ds (cid:0) ∂ ik ( m i (1 − m k )) (cid:1) ( sg ik ) (cid:19) ≤ C E X k,l = i | g ik | | g il | ≤ CN as well as E (cid:18) X k = i g ik Z ds Z s ds (cid:0) ∂ ik m k (cid:1) ( s g ik ) (cid:19) ≤ C E X k,l = i | g ik | | g il | ≤ CN . With the identities ∂ il m k = m i m kl + m l m ik + m ilk ,∂ il m ik = (1 − m i ) m kl − m il m ik − m i m l m ik − m i m ilk , (5.2)we then obtain by Gaussian integration by parts E (cid:18) X k = i g ik W k ( g ik = 0) (cid:19) = tN E X k = i W k ( g ik = 0) + t N E X k,l = i ( ∂ il m k ( g ik = 0))( ∂ ik m l ( g il = 0)) . Here, we used that ∂ ik (cid:0) W k ( g ik = 0) (cid:1) = 0 and that ∂ il m ( i ) k = 0 (for all k, l ∈ { , . . . , N } ).Now, Eq. (5.2) and Lemma 3.1 together with the remarks thereafter show that t N (cid:12)(cid:12)(cid:12)(cid:12) E X k,l = i ( ∂ il m k ( g ik = 0))( ∂ ik m l ( g il = 0)) (cid:12)(cid:12)(cid:12)(cid:12) ≤ t N E X k,l = i (cid:0) m i m kl + m l m ik + m ilk (cid:1) ( g ik = 0) ≤ CN . Notice that applying Lemma 3.1 is enough to obtain the previous bound, because we canbound the L ( P ) norms of the three point functions m ikl by the L ( P ) norms of suitabletwo point functions, through the identity (3.2).19o estimate tN − E P k = i W k ( g ik = 0), on the other hand, we recall Eq. (4.3) so that W k ( g ik = 0) = (cid:16)(cid:10) m [ i ] k (cid:11) − m ( i ) k (cid:17) ( g ik = 0)= X j = i,k (cid:18) Z t (cid:10) σ i m [ i ] jk (cid:11) ( s ) dg ij ( s ) − Z t (cid:10) m [ i ] j m [ i ] jk (cid:11) ( s ) dsN (cid:19) ( g ik = 0) . Hence, Lemma 3.1 implies also in this case that tN E X k = i W k ( g ik = 0) ≤ CN and, similarly, for the remaining contribution that E (cid:18) X k = i g ik ( m k m ik )( g ik = 0) (cid:19) ≤ E (cid:18) X k = i g ik (cid:19)(cid:18) X k = i m ik ( g ik = 0) (cid:19) = E (cid:18) X k = i g ik (cid:19) E (cid:18) X k = i m ik ( g ik = 0) (cid:19) ≤ CN . Collecting the previous bounds, we conclude that X k = i g ik m ( i ) k = X k = i g ik m k − X k = i g ik (1 − m k ) m i − X k = i g ik X k = X k = i g ik m k − t (1 − q N ) m i − X k = i (cid:0) g ik − t/N (cid:1) (1 − m k ) m i − X k = i g ik X k . (5.3)Here, the error term P k = i g ik X k satisfies the estimate (5.1) and, arguing once more asabove, we also find that E (cid:18) X k = i (cid:0) g ik − t/N (cid:1) (1 − m k ) m i (cid:19) = E X k,l = i : k = l (cid:0) g ik g il − g ik t/N + t /N (cid:1) (1 − m k )(1 − m l ) m i + E X k = i (cid:0) g ik − g ik t/N + t /N (cid:1) (1 − m k ) m i ≤ E tN X k,l = i : k = l (cid:0) g ik g il − g ik t/N (cid:1) ∂ ik h (1 − m k )(1 − m l ) m i i + E t N X k,l = i : k = l g il ∂ il h (1 − m k )(1 − m l ) m i i + CN ≤ CN . Note that the last bound follows from repeated Gaussian integration by parts and thefact that derivatives of (1 − m k ) m i are bounded by some constant C > 0. By the20ipschitz continuity of x tanh( x ), this proves with Eq. (5.3) the TAP equations(1.10).Let us now turn to the proof of the TAP equations (1.11) for the two point functions.We use the same ideas as for the proof of Eq. (1.10) and focus on the main steps. ByLemma 1.5, we have to determine the leading order contribution to X k = i g ik (cid:0) m kj − m ( i ) kj (cid:1) =: X k = i g ik Y k . We view Y k = Y k ( g ik ) as a function of g ik and a second order Taylor expansion yields Y k ( g ik ) = Y k ( g ik = 0) + g ik (cid:0) (1 − m k ) m ij − m i m k m kj (cid:1) ( g ik = 0) − g ik (cid:0) m ik m jk + m k m ijk (cid:1) ( g ik = 0) + g ik Z ds Z s ds (cid:0) ∂ ik m kj (cid:1) ( s g ik ) . Hence, defining Z k := Y k − g ik (1 − m k ) m ij + 2 g ik m i m k m kj , we find Z k = Y k ( g ik = 0) − g ik Z ds ∂ ik (cid:0) (1 − m k ) m ij − m i m k m kj (cid:1) ( sg ik ) − g ik (cid:0) m ik m jk + m k m ijk (cid:1) ( g ik = 0) + g ik Z ds Z s ds (cid:0) ∂ ik m kj (cid:1) ( s g ik ) . (5.4)Now, in the first step, we prove that for all ǫ > E (cid:18) X k = i g ik Z k (cid:19) ≤ CN ǫ . (5.5)This follows from the decay results of Lemma 3.1, 3.2 and the remarks following theirproofs. We start with the term E (cid:18) X k = i g ik Z ds ∂ ik (cid:0) (1 − m k ) m ij − m i m k m kj (cid:1) ( sg ik ) (cid:19) ≤ C sup s ∈ [0;1] E X k,l = i | g ik || g il || m ij ( sg ik ) | + C sup s ∈ [0;1] E X k,l = i,j | g ik || g il || m kj ( sg ik ) | + CN + C sup s ∈ [0;1] E X k,l = i,j | g ik || g il || ∂ ik m ij ( sg ik ) | + C sup s ∈ [0;1] E X k,l = i,j | g ik || g il || ∂ ik m kj ( sg ik ) | + CN ≤ CN / , where we recall that we assume i = j and where we used the identity ∂ ik m kj = − m j (cid:0) m i m jk + m k m ij + m ijk (cid:1)(cid:0) δ j m [ j ] k (cid:1) + (1 − m j ) δ j h(cid:0) − ( m [ j ] k ) (cid:1) m [ j ] i − m [ j ] k m [ j ] ik i (5.6)21o control the terms involving ∂ ik m ij and ∂ ik m kj . Observe that Eq. (5.6) is a simpleconsequence of the conditional identity (3.1). Notice also that, here and in the following,we frequently use rough bounds of the form E m ij ≤ C E m ij so that all of the followingbounds hold true for times t < log 2.Analogously to the last bound, we obtain that E (cid:18) X k = i g ik (cid:0) m ik m jk + m k m ijk (cid:1) ( g ik = 0) (cid:19) ≤ C X k,l = i,j E g ik E (cid:0) m ik m jk (cid:1) ( g ik = 0) + C X k,l = i,j E g ik E (cid:0) m ijk (cid:1) ( g ik = 0) + CN ≤ CN ǫ . To bound the last contribution on the right hand side of Eq. (5.4), we differentiatethe identity (5.6) and a tedious, but straight forward computation shows that ∂ ik m kj = − (cid:0) m i m jk + m k m ij + m ijk (cid:1) (cid:0) δ j m [ j ] k (cid:1) − m j h ∂ ik (cid:0) m i m jk + m k m ij + m ijk (cid:1)i(cid:0) δ j m [ j ] k (cid:1) − m j (cid:0) m i m jk + m k m ij + m ijk (cid:1)h ∂ ik (cid:0) δ j m [ j ] k (cid:1)i − m j (cid:0) m i m jk + m k m ij + m ijk (cid:1) δ j h(cid:0) − ( m [ j ] k ) (cid:1) m [ j ] i − m [ j ] k m [ j ] ik i + (1 − m j ) δ j h − m [ j ] k (cid:0) − ( m [ j ] k ) (cid:1)(cid:0) m [ j ] i (cid:1) − m [ j ] i (cid:0) m [ j ] k (cid:1) m [ j ] ik i − (1 − m j ) δ j h(cid:0) − ( m [ j ] k ) (cid:1) m [ j ] i + 2 m [ j ] k ( m [ j ] ik ) + (cid:0) − ( m [ j ] k ) (cid:1) m [ j ] i m [ j ] ik i . If we then proceed as above, using the bounds from Lemmas 3.1 and 3.2 combined withthe product rule for the action of δ j (in the last formula), we verify that E (cid:18) X k = i g ik Z ds Z s ds (cid:0) ∂ ik m kj (cid:1) ( s g ik ) (cid:19) ≤ C sup s ∈ [0;1] E X k,l = i,j g ik (cid:0) ∂ ik m kj (cid:1) ( sg ik ) + CN ≤ CN / . Finally, it remains to bound the first term on the right hand side in Eq. (5.4). We have E (cid:18) X k = i g ik Y k ( g ik = 0) (cid:19) = E tN X k = i Y k ( g ik = 0) + E t N X k,l = i : k = l ( ∂ ik m lj )( g il = 0)( ∂ il m kj )( g ik = 0)+ E t N X k = i ( ∂ ik m kj ) ( g ik = 0) ≤ E tN X k = i Y k ( g ik = 0) + E t N X k,l = i : k = l ( ∂ ik m lj ) ( g il = 0) + CN , E tN X k = i Y k ( g ik = 0) ≤ CN ǫ and the smallness of the last contribution follows from the identity ∂ ik m lj = − m l (cid:0) m i m kl + m k m il + m ilk (cid:1)(cid:0) δ l m [ l ] j (cid:1) + (1 − m l ) δ l (cid:0) m [ l ] i m [ l ] kj + m [ l ] k m [ l ] ij + m [ l ] ijk (cid:1) . (5.7)It implies with the product rule for δ l and the identity (3.2) that E t N X k,l = i : k = l ( ∂ ik m lj ) ( g il = 0) ≤ CN ǫ . Collecting the previous estimates, we summarize that we have shown that X k = i g ik m ( i ) kj = X k = i g ik m kj + 2 X k = i g ik m jk m k m i − X k = i g ik (1 − m k ) m ij − X k = i g ik Z k , where the error P k = i g ik Z k satisfies the estimate (5.5). To conclude the TAP equations(1.11), it thus only remains to replace g ik by its mean in the previous equation and toshow that the resulting error is small. To this end, we apply once more the argumentsfrom the previous steps to deduce that E (cid:18) X k = i (cid:0) g ik − t/N (cid:1) m jk m k m i (cid:19) + E (cid:18) X k = i (cid:0) g ik − t/N (cid:1) (1 − m k ) m ij (cid:19) ≤ CN ǫ . We omit the details and conclude the proof of Corollary (1.4). E m ij In this section, we outline the proofs of Propositions 1.2 and 1.3. Let us start with theproof of the concentration of the overlap, Eq. (1.7). This is a consequence of the TAPequations (1.4) for the magnetizations m i and follows from a contraction argument. Proof of Proposition 1.2. Let Z ∼ N (0 , 1) denote a standard Gaussian random variable,independent of the disorder ( g ij ) ≤ i Since the last two estimates can be proved with the same arguments as in Sections 3and 4, we skip the details. What they imply is that (cid:12)(cid:12)(cid:12) E q N − E tanh (cid:16) h + X k =1 g k m (1) k (cid:17)(cid:12)(cid:12)(cid:12) ≤ CN / , (cid:12)(cid:12)(cid:12) E q N − E tanh (cid:16) h + X k =1 , g k m (1 , k (cid:17) tanh (cid:16) h + X k =1 , g k m (1 , k (cid:17)(cid:12)(cid:12)(cid:12) ≤ CN / . q (1) N = N − P k =1 (cid:0) m (1) k (cid:1) , we have (as observed in [22, Lemma 1.7.6]) that Z := (cid:0) tq (1) N (cid:1) − / X k =1 g k m (1) k ∼ N (0 , g kl for all k, l = 1 (and hence unconditionally Gaussian). Therefore E tanh (cid:16) h + X k =1 g k m (1) k (cid:17) = E f (cid:0) q (1) N (cid:1) . Similarly, defining q (1 , N = N − P k =1 , (cid:0) m (1 , k (cid:1) as well as the Gaussians Z := (cid:0) tq (1 , N (cid:1) − / X k =1 , g k m (1 , k ∼ N (0 , ,Z := (cid:0) tq (1 , N (cid:1) − / X k =1 , g k m (1 , k ∼ N (0 , , we easily see that E g • g • Z = 1 , E g • g • Z = 1 , E g • g • Z Z = 0 . Here, E g • g • denotes the expectation conditionally on g kl for all k, l = 1 , 2. Thus, Z and Z are, conditionally on g kl for all k, l = 1 , 2, two i.i.d. standard Gaussians. Sincetheir conditional statistics is deterministic, ( Z , Z ) ∼ N (0 , R ) is unconditionallyjointly Gaussian, and independent of the remaining disorder g kl for all k, l = 1 , E tanh (cid:16) h + X k =1 , g k m (1 , k (cid:17) tanh (cid:16) h + X k =1 , g k m (1 , k (cid:17) = E E g • g • tanh (cid:16) h + q tq (1 , N Z (cid:17) tanh (cid:16) h + q tq (1 , N Z (cid:17) = E f (cid:0) q (1 , N (cid:1) Finally, let us point out that the Lipschitz continuity of f implies that (cid:12)(cid:12) E f (cid:0) q (1) N (cid:1) − E f (cid:0) q N (cid:1)(cid:12)(cid:12) ≤ (cid:13)(cid:13) m − m (1)2 (cid:13)(cid:13) + CN / ≤ CN / , (cid:12)(cid:12) E f (cid:0) q (1 , N (cid:1) − E f (cid:0) q N (cid:1)(cid:12)(cid:12) ≤ (cid:13)(cid:13) m − m (1)3 (cid:13)(cid:13) + 2 (cid:13)(cid:13) m (1)3 − m (1 , (cid:13)(cid:13) + CN / ≤ CN / . Collecting the above observations, we obtain that E (cid:12)(cid:12) q N − E q N (cid:12)(cid:12) ≤ E (cid:0) q N − f ( E q N ) (cid:1) = E q N − f ( E q N ) E q N + f ( E q N ) ≤ E f ( q N ) − f ( E q N ) E f ( q N ) + f ( E q N ) + CN / = E (cid:12)(cid:12) f ( q N ) − f ( E q N ) (cid:12)(cid:12) + CN / ≤ sup x ∈ [0; ∞ ) | f ′ ( x ) | E (cid:12)(cid:12) q N − E q N (cid:12)(cid:12) + CN / . x ∈ [0; ∞ ) | f ′ ( x ) | ≤ t < 1, this proves that q N concentrates, i.e. E (cid:12)(cid:12) q N − E q N (cid:12)(cid:12) ≤ CN / . Using again the Lipschitz continuity of f , it also shows that (cid:12)(cid:12) E q N − f (cid:0) E q N (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) E f ( q N ) − f (cid:0) E q N (cid:1)(cid:12)(cid:12) + CN / ≤ CN / . If q ∈ [0; 1] denotes the unique fixed point q = E Z tanh ( h + √ tqZ ) = f ( q ) (for theuniqueness, see for instance [22, Prop. 1.3.8] and recall that t < (cid:12)(cid:12) q − E q N (cid:12)(cid:12) ≤ (cid:12)(cid:12) f ( q ) − f ( E q N ) (cid:12)(cid:12) + CN / ≤ t (cid:12)(cid:12) q − E q N (cid:12)(cid:12) + CN / , so that (cid:12)(cid:12) q − E q N (cid:12)(cid:12) ≤ C/N / . This implies in particular (1.7) and finishes the proof.Having proved the concentration of the overlap, let us now make the heuristics (1.8)rigorous in order to prove Proposition 1.3. Before we start, we record that E (cid:12)(cid:12) q − q N (cid:12)(cid:12) p ≤ CN / (6.1)for any p ≥ 2, which follows by interpolation from the concentration bound (1.7) andthe boundedness of q N = N − P Nk =1 m k ≤ Proof of Proposition 1.3. By the TAP equations (1.5) and Gaussian integration by parts,we find that E m ij = E t sech (cid:16) h + X k = i g ik m ( i ) k (cid:17) N X l = i (cid:0) m ( i ) lj (cid:1) + E t N (cid:18) X l = i m ( i ) l m ( i ) lj (cid:19) (cid:0) (cid:0) h + P k = i g ik m ( i ) k (cid:1) − (cid:1) cosh (cid:0) h + P k = i g ik m ( i ) k (cid:1) + Θ , where the error Θ satisfies | Θ | ≤ C/N ǫ , for ǫ > x ∈ R (cid:12)(cid:12) ( x ) − ( x ) (cid:12)(cid:12) ≤ C so that (cid:12)(cid:12)(cid:12)(cid:12) E t N (cid:18) X l = i m ( i ) l m ( i ) lj (cid:19) (cid:0) (cid:0) h + P k = i g ik m ( i ) k (cid:1) − (cid:1) cosh (cid:0) h + P k = i g ik m ( i ) k (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ E t N (cid:18) X l = i m ( i ) l m ( i ) lj (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) (cid:0) (cid:0) h + P k = i g ik m ( i ) k (cid:1) − (cid:1) cosh (cid:0) h + P k = i g ik m ( i ) k (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C E (cid:18) N X l = i,j m ( i ) l m ( i ) lj (cid:19) + CN / , (6.2)26here the last bound follows from Lemma 3.1.To continue, we control the first term on the right hand side of the last equationthrough another contraction argument. This term is an expectation over mixed corre-lation functions and we are going to show that this term is of lower order o ( N − ), asclaimed in (1.8). To make this rigorous, it is first of all useful to observe that E (cid:18) N X l = i,j h m l m lj − m ( i ) l m ( i ) lj i(cid:19) ≤ CN ǫ . (6.3)This can be proved using the results of Lemma 3.1 and 3.2, proceeding as in Section 4(recall in particular Eq. (4.6)); we omit the details. By Lemma 4.2, we then see that E (cid:18) N X l = i,j m l m lj (cid:19) = E (cid:18) N X l = i,j m l (1 − m j ) X k = j g jk m ( j ) kl (cid:19) + Θ with an error Θ such that | Θ | ≤ C/N ǫ . Since we can pull the non–negative factor(1 − m j ) ≤ E (cid:18) N X l = i,j m l (1 − m j ) X k = j g jk m ( j ) kl (cid:19) ≤ E (cid:18) N X l = i,j m l X k = j g jk m ( j ) kl (cid:19) = E tN X l ,l = i,j ; k = j m l m l m ( j ) kl m ( j ) kl + E t N X l ,l = i,j ; k ,k = j m ( j ) k l m ( j ) k l ∂ jk ∂ jk (cid:0) m l m l (cid:1) = E tN (cid:18) X l = i,j,r m l m ( j ) lr (cid:19) + E t N X l ,l = i,j ; k ,k = j,l ,l ; k = k ,l = l m ( j ) k l m ( j ) k l ∂ jk ∂ jk (cid:0) m l m l (cid:1) + Θ for an error Θ such that | Θ | ≤ C/N ǫ and some fixed r = j , by symmetry. But then,on the one hand, we can use Eq. (5.2), the identity (3.2) and Eq. (5.7) to deduce that E t N X l ,l = i,j ; k ,k = j,l ,l ; k = k ,l = l m ( j ) k l m ( j ) k l ∂ jk ∂ jk (cid:0) m l m l (cid:1) = E t N X l ,l = i,j ; k ,k = j,l ,l ; k = k ,l = l m ( j ) k l m ( j ) k l ∂ jk h m l (cid:0) m j m k l + m k m jl + m jk l (cid:1)i ≤ CN ǫ . On the other hand, we find with similar arguments as before that E (cid:18) N X l = i,j,r m l (cid:0) m lr − m ( j ) lr (cid:1)(cid:19) ≤ CN ǫ . E (cid:18) N X l = i,j m l m lj (cid:19) ≤ t E (cid:18) N X l = i,j,r m l m lr (cid:19) + CN ǫ ≤ t E (cid:18) N X l = i,j m l m lj (cid:19) + CN ǫ , and we conclude under the assumption t < log 2 < E (cid:18) N X l = i,j m l m lj (cid:19) ≤ C (1 − t ) − N ǫ ≤ CN ǫ . By Eq. (6.3), this also implies that E (cid:18) N X l = i,j m ( i ) l m ( i ) lj (cid:19) ≤ CN ǫ and plugging this into Eq. (6.2), it follows that (cid:12)(cid:12)(cid:12)(cid:12) E t N (cid:18) X l = i m ( i ) l m ( i ) lj (cid:19) (cid:0) (cid:0) h + P k = i g ik m ( i ) k (cid:1) − (cid:1) cosh (cid:0) h + P k = i g ik m ( i ) k (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CN ǫ . This proves E m ij = E t sech (cid:16) h + X k = i g ik m ( i ) k (cid:17) N X l = i (cid:0) m ( i ) lj (cid:1) + Θ , for an error | Θ | ≤ C/N ǫ , for any ǫ > q ( i ) N (recall that q ( i ) N and q N are close in L ( P )) implies that E m ij = E t sech (cid:16) h + X k = i g ik m ( i ) k (cid:17) N X l = i (cid:0) m ( i ) lj (cid:1) + Θ = E t sech (cid:0) h + √ tqZ (cid:1) E N X l = i (cid:0) m ( i ) lj (cid:1) + Θ for Z ∼ N (0 , 1) independent of the remaining disorder and an error | Θ | ≤ C/N ǫ .Here, we have used the Lipschitz continuity of the map x E Z sech (cid:16) h + √ txZ (cid:17) and, choosing δ > E E Z | m ( i ) lj | h sech (cid:16) h + q tq ( i ) N Z (cid:17) − sech (cid:16) h + √ tqZ (cid:17)i ≤ C (cid:13)(cid:13) m ( i ) lj (cid:13)(cid:13) δ (cid:13)(cid:13) q ( i ) N − q k δ/ (2+ δ )(2+ δ ) /δ ≤ CN ǫ ǫ = ǫ δ > q ( i ) N ).Replacing then the m ( i ) lj by m lj through Itˆo’s Lemma as in Section 4 and usingsymmetry over the sites shows that E m ij = 1 N E t sech (cid:0) h + √ tqZ (cid:1) E (1 − m j ) + E t sech (cid:0) h + √ tqZ (cid:1) E m ij + Θ for an error | Θ | ≤ C/N ǫ . 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