We provide a simple extension of Bolthausen’s Morita-type proof of the replica symmetric formula [E. Bolthausen, “A Morita type proof of the replica-symmetric formula for SK,” in Statistical Mechanics of Classical and Disordered Systems, Springer Proceedings in Mathematics and Statistics (Springer, Cham., 2018), pp. 63–93; arXiv:1809.07972] for the Sherrington–Kirkpatrick model and prove the replica symmetry for all (β, h) that satisfy β2Esech2(βqZ+h)1, where q=Etanh2(βqZ+h). Compared to the work of Bolthausen [“A Morita type proof of the replica-symmetric formula for SK,” in Statistical Mechanics of Classical and Disordered Systems, Springer Proceedings in Mathematics and Statistics (Springer, Cham., 2018), pp. 63–93; arXiv:1809.07972], the key of the argument is to apply the conditional second moment method to a suitably reduced partition function.

We study systems of N spins σi, i ∈ {1, …, N}, with values in {−1, 1} and with the Hamiltonian HN:{1,1}NR defined by

HN(σ)=β21i,jNgijσiσj+hi=1Nσi.
(1.1)

The interactions {gij} are i.i.d. centered Gaussians of variance 1/N for ij, and we set gii ≡ 0. β ≥ 0 denotes the inverse temperature, and h > 0 denotes the external field strength.

Equation (1.1) corresponds to the Sherrington–Kirkpatrick (SK) spin glass model,1 and we are interested in its free energy fN at high temperature, where

fN=1NlogZN,ZN=σ{1,1}NeHN(σ).
(1.2)

The mathematical understanding of the SK model has required substantial efforts until the famous Parisi formula2,3 was rigorously established by Guerra4 and Talagrand.5 Later, Panchenko6,7 gave another proof based on the ultrametricity of generic models. For a thorough introduction to the topic, we refer to Refs. 8–11.

Despite the validity of the Parisi formula, it is an interesting question to prove the replica symmetry of the SK model at high temperature, as predicted by de Almeida and Thouless.12 Replica symmetry is expected for all (β, h) that satisfy

β2E sech4(βqZ+h)<1,
(1.3)

where q denotes the unique solution of the self-consistent equation,

q=Etanh2(βqZ+h).
(1.4)

In both cases, ZN(0,1) denotes a standard Gaussian and E denotes the expectation over Z.

In the special case that the external field in the direction of σi is a centered Gaussian random variable, hi = hgi, for i.i.d. giN(0,1) (independent of gij), replica symmetry has recently been shown in Ref. 13 for all (β, h) that satisfy (1.3) [in which case h in (1.3) and (1.4) is replaced by hZ′ for some ZN(0,1) independent of Z]; see also Ref. 14 for previous results in this case. For Hamiltonians HN as in (1.1) (or, more generally, Hamiltonians with a non-centered random external field), however, replica symmetry is to date only known above the AT line up to a bounded region in the (β, h)-phase diagram. This was analyzed in Ref. 15. Like Ref. 13, this analysis is based on the Parisi variational problem and we refer to Ref. 15 for the details. For previously obtained results based on the Parisi formula, see also Ref. 11, Chap. 13.

In this article, instead of analyzing the high temperature regime in view of the Parisi variational problem, we give a simple extension of Bolthausen’s argument16 and prove the replica symmetric formula for all (β, h) that satisfy

β2E sech2(βqZ+h)1.
(1.5)

Although (1.5) is clearly stronger than condition (1.3), it already covers a fairly large region of the high temperature regime; see Fig. 1 for a schematic. It improves upon the inverse temperature range from Ref. 16, where β was assumed to be sufficiently small.

FIG. 1.

Schematic of the (T, h) phase diagram, where T=1β denotes the temperature. In the blue region, whose boundary corresponds to the AT line (1.3), the SK model is known to be replica symmetry breaking. The boundary of the green region corresponds to condition (1.5). Theorem 1.1 proves the replica symmetry in the green region.

FIG. 1.

Schematic of the (T, h) phase diagram, where T=1β denotes the temperature. In the blue region, whose boundary corresponds to the AT line (1.3), the SK model is known to be replica symmetry breaking. The boundary of the green region corresponds to condition (1.5). Theorem 1.1 proves the replica symmetry in the green region.

Close modal

Theorem 1.1.
Assume that (β, h) satisfies(1.5). Then,
limNE1NlogZN=log2+Elogcosh(βqZ+h)+β24(1q)2.
(1.6)

Remarks.

  1. From Ref. 17, it is well-known that limNE1NlogZN exists and that almost surely limN1NlogZN=limNE1NlogZN.

  2. It follows from the results of Ref. 4 that the right-hand side of (1.6) provides an upper bound to the free energy limNE1NlogZN for all inverse temperatures and external fields (β, h). To establish Theorem 1.1, it is, therefore, sufficient to prove that the right-hand side of (1.6) provides a lower bound to limNE1NlogZN.

We conclude the Introduction with a quick heuristic outline of the main argument. To this end, consider first the case h = 0 where the critical temperature corresponds to β = 1. In this case, it is straightforward (see, e.g., Ref. 18, Chap. 1, Sec. 3) to see that
limN1NlogEZN=log2+β24,limN1NlogEZN2=2log2+β22
for all β < 1. The replica symmetric formula, thus, follows from the second moment method using the Gaussian concentration of the free energy. In fact, for h = 0, the fluctuations of ZN/EZN have also been known for a long time.19 
Clearly, it would be desirable to extend this simple argument to the case h > 0, but a direct application of the second moment method does not work here. However, as suggested in Ref. 16, one may hope to obtain a model similar to the case h = 0 by centering the spins around suitable magnetizations and viewing ZN, up to normalization, as an average over the corresponding coin-tossing measure. To center the spins correctly, recall that at high temperature, one expects the TAP equations20 to hold, that is,
mitanhh+βjiḡijmjβ2(1q)mi(fori=1,,N),
(1.7)
where ḡij=(gij+gji)/2 and mi=ZN1σσieHN(σ). The validity of (1.7) is known for sufficiently small β (see Refs. 10 and 21 and, more recently, Ref. 22; see also Ref. 23 on the TAP equations for generic models, valid at all temperatures) and expected to be true under (1.3). In Refs. 16 and 24, Bolthausen provided an iterative construction (m(s))sN of the solution to (1.7) that converges (in a suitable sense) in the full high temperature regime (1.3). The main result of Ref. 16 is a novel proof of (1.6) for β small enough based on a conditional second moment argument, given the approximate solutions (m(s))sN. It has remained an open question, however, if the approach can be extended to the region (1.3).
In this article, while we are not able to resolve this question for all (β, h) satisfying (1.3), we improve the range of (β, h) to (1.5) as follows: The author of Refs. 16 and 24 showed, roughly speaking, that m(k+1)s=1kγsϕ(s) for certain orthonormal vectors ϕ(s)RN and deterministic numbers γs ( ≈ ⟨m(k+1), ϕ(s)⟩ with high probability), where x,y=N1i=1Nxiyi for x,yRN. One also has g=g(k+1)+s=1kρ(s) for the interaction g=(gij)1i,jN, where ρ(s)RN×N are measurable with respect to (m(s))sk+1 and where the modified interaction g(k+1) is Gaussian, conditionally on (m(s))sk+1, with the property that g(k+1)s=1kγsϕ(s)=0. Up to negligible errors, one obtains with σ̂=σm(k+1) that
1NlogZNlog2+Elogcosh(βqZ+h)+1Nlogσ{1,1}Npfree(σ)expNβ2σ̂,g(k)σ̂+NOmaxs|γsσ,ϕ(s)|,
(1.8)
where pfree denotes the product measure for which ∑σpfree(σ) σ = m(k+1). A simple observation is now that we can ignore the error NO(maxs|γsσ,ϕ(s)|) in (1.8) by restricting the modified partition function to those σ with maxs|γs − ⟨σ, ϕ(s)⟩| ≈ 0. Note that the probability of the complement of this set is small under pfree because γs ≈ ⟨m(k+1), ϕ(s)⟩. This yields a simple lower bound on 1NlogZN, and we can apply the conditional second moment argument to the restricted partition function. We show that its first conditional moment equals β2(1 − q)2/4 (up to negligible errors) in the full high temperature regime (1.3). To dominate its second moment by the square of the first, on the other hand, we need to impose the stronger condition (1.5).

Note that imposing similar orthogonality restrictions on the partition function has been proved useful before for obtaining lower bounds on the free energy, such as in the TAP analysis of the spherical SK model.25 

Although (1.5) already covers a comparably large region of the high temperature phase, as schematically shown in Fig. 1, it remains an open question whether the second moment argument can be extended to the full high temperature regime (1.3); see also the related comments in Ref. 16, Sec. 6.

This paper is structured as follows. In Secs. III and IV, we set up the notation and recall Bolthausen’s iterative construction of magnetization.16,24 In Sec. IV, we define the reduced partition function and compute its first and second moments. In Sec. V, we apply the conditional second moment method to prove Theorem 1.1.

In this section, we introduce the basic notation and conventions. We closely follow Ref. 16.

We usually denote vectors in RN by boldface or greek letters. If xRN and g:RR, we define g(x) in the component-wise sense. By ,:RN×RNR, we denote the normalized inner product

x,y=1Ni=1Nxiyi,

and by =,, we denote the induced norm. We also normalize the tensor product xy:RNRN of two vectors x,yRN so that for all zRN,

(xy)(z)=y,zx.

Given a matrix ARN×N, ATRN×N denotes its transpose and ĀRN×N denotes its symmetrization,

Ā=A+AT2.

We mostly use the letters Z, Z′, Z1, Z2, etc. to denote standard Gaussian random variables independent of the disorder {gij} and independent of one another. When we average over such Gaussians, we denote the corresponding expectation by E to distinguish it from the expectation E with respect to the disorder {gij}. Unless specified otherwise, we consider all Gaussians to be centered.

Finally, given two sequences of random variables (XN)N1,(YN)N1 that may depend on parameters such as β and h, we say that

XNYN

if and only if there exist positive constants c, C > 0, which may depend on the parameters, but which are independent of N, such that for every t > 0, we have

P(|XNYN|>t)Cect2N.

In this section, we recall Bolthausen’s iterative construction of the solution to the TAP equations16,24 and list the properties that we will need for the Proof of Theorem 1.1. We follow here the conventions of Ref. 16, and we refer to Ref. 16, Secs. 2, 4, and 5 for the proofs of the following statements.

First of all, we define three sequences (αk)kN, (γk)kN, and (Γk)kN. Set

α1=qγ1,γ1=Etanh(βqZ+h),Γ12=γ12,

where here and in the following, q denotes the unique solution of (1.4). Then, we define ψ : [0, q] → [0, q] by

ψ(t)=Etanh(βtZ+βqtZ+h)tanh(βtZ+βqtZ+h),

and set recursively

αk=ψ(αk1),γk=αkΓk12qΓk12,Γk2=j=1kγk2.

The following lemma collects important properties of (αk)kN, (γk)kN, and (Γk)kN.

Lemma 3.1.

(Ref.24 , Lemma 2.2, Corollary 2.3, Lemma 2.4 and Ref.16 , Lemma 2)

  1. ψis strictly increasing and convex in [0, q] with 0 < ψ(0) < ψ(q) = q. If(1.3)is satisfied, thenqis the unique fixed point ofψin [0, q].

  2. The sequence(αk)kNis increasing andαk > 0 for everykN. If(1.3)is satisfied, thenlimkαk=q, and if(1.3)is satisfied with a strict inequality, the convergence is exponentially fast.

  3. For allk ≥ 2, we have that0<Γk12<αk<qand that0<γk<qΓk12. If(1.3)is true, thenlimkΓk2=qand, as a consequence,limkγk=0.

Next, we recall Bolthausen’s modified interaction matrix. We define

g(1)=g,ϕ(1)=1RN,m(1)=q1RN.

Assuming that g(s), ϕ(s), m(s) are defined for 1 ≤ sk, we set

ζ(s)=ḡ(s)ϕ(s)

and we define the σ-algebra Gk through

Gk=σg(s)ϕ(s),(g(s))Tϕ(s):1sk.

Expectations with respect to Gk are denoted by Ek. Furthermore, we set

h(k+1)=h1+βs=1k1γsζ(s)+βqΓk12ζ(k),m(k+1)=tanh(h(k+1)),ϕ(k+1)=m(k+1)s=1km(k+1),ϕ(s)ϕ(s)m(k+1)s=1km(k+1),ϕ(s)ϕ(s),
(3.1)

and we note that ϕ(k+1) is Pa.s well-defined for all k if k < N (Ref. 16, Lemma 5). Finally, the modified interaction matrix g(k+1) is defined by

g(k+1)=g(k)ρ(k),

where

ρ(k)=g(k)ϕ(k)ϕ(k)+ϕ(k)(g(k))Tϕ(k)g(k)ϕ(s),ϕ(k)ϕ(k)ϕ(k).

In particular, this means that ḡ(k+1) is equal to

ḡ(k+1)=ḡ(k)ρ̄(k),ρ̄(k)=ζ(k)ϕ(k)+ϕ(k)ζ(k)ζ(k),ϕ(k)ϕ(k)ϕ(k).

It is clear that (ϕ(s))s=1k forms an orthonormal sequence of vectors in RN, and we denote by P(k) and Q(k) the corresponding orthogonal projections in RN, that is,

P(k)=s=1kϕ(s)ϕ(s)=(Pij(k))1i,jN,Q(k)=1P(k)=(Qij(k))1i,jN.

By Ref. 16, Lemma 3, m(k) and ϕ(k) are Gk1-measurable for all kN, and we also have that

g(k)ϕ(s)=(g(k))Tϕ(s)=ḡ(k)ϕ(s)=0,s<k.

Proposition 3.2.

(Ref.16 , Proposition 4)

  1. Conditionally onGk2,g(k)andg(k−1)are Gaussian with conditional covariance, givenGk2, equal to
    Ek2gij(k)gst(k)=1NQis(k1)Qjt(k1).
  2. Conditionally onGk2,g(k)is independent ofGk1. In particular, conditionally onGk1,g(k)is Gaussian with the same covarianceas in (1).

  3. Conditionally onGk1, the random variablesζ(k)are Gaussian with
    Ek1ζi(k)ζj(k)=Qij(k1)+1Nϕi(k)ϕj(k).

The main result of Ref. 24 is summarized in the following proposition.

Proposition 3.3.
(Ref.24 , Proposition 2.5 and Ref.16 , Proposition 6) For everykNands < k, one has
m(k),ϕ(s)γs,m(k),m(s)αs,m(k),m(k)q.

The next lemma collects a few auxiliary results that are helpful in the sequel.

Lemma 3.4.

[Ref.16 , Lemmas 11, 14, and 15(b)]

  1. For everykN,ϕ(k),ζ(k)=2ϕ(k),g(k)ϕ(k)is unconditionally Gaussian with variance 2/N.

  2. For every Lipschitz continuousf:RRwith |f(x)| ≤ C(1 + |x|) for someC > 0, one has for allk ≥ 2,
    limNE1Ni=1Nf(hi(k+1))Ef(βqZ+h)=0.
  3. For everykNandt > 0, it holds true that
    limNPζ(k)1+t=0.

Using Bolthausen’s magnetizations, we compute in this section the first two conditional moments of a suitably reduced partition function onto a subset S ⊂ {−1,1}N defined below. This will suffice to establish Theorem 1.1, as explained in Sec. V.

Let ɛ > 0 and kN be fixed. We define the set Sɛ,k ⊂ {−1,1}N through

Sε,k=σ{1,1}N:|σm(k+1),ϕ(s)|ε/k,1sk
(4.1)

with ϕ(s), γs from Sec. III. We define the reduced partition function ZN(k+1)(Sε,k) by

ZN(k+1)(Sε,k)=σSε,kpfree(σ)expNβ2σ,g(k+1)σ=σSε,kpfree(σ)expNβ2σ,ḡ(k+1)σ,
(4.2)

where pfree : {−1,1}N → (0, 1) denotes the coin-tossing measure,

pfree(σ)=i=1N12exphi(k+1)σicoshhi(k+1).
(4.3)

The following lemma records that pfree(Sε,kc) is exponentially small in N.

Lemma 4.1.
Letɛ > 0,kN, and letSɛ,kandpfreebe defined as in(4.1)and(4.3), respectively. Then, there existc, C > 0, independent ofNandɛ, such that
pfree(Sε,k)1CecNε2.
(4.4)

Proof.
By a standard union bound, we have that
pfree(Sε,kc)kmaxs=1,,kpfreeσ{1,1}N:σ,ϕ(s)m(k+1),ϕ(s)>εk+kmaxs=1,,kpfreeσ{1,1}N:m(k+1),ϕ(s)σ,ϕ(s)>εk,
which implies that
pfree(Sε,kc)kmaxs=1,,kinfλ0expNλεk1Ni=1Nlogcoshhi(k+1)+λϕi(s)coshhi(k+1)+λm(k+1),ϕ(s)+kmaxs=1,,kinfλ0expNλεk1Ni=1Nlogcoshhi(k+1)λϕi(s)coshhi(k+1)λm(k+1),ϕ(s).
Using the pointwise bound logcosh(x+y)logcosh(x)+ytanh(x)+y22 for x,yR, ⟨ϕ(s), ϕ(s)⟩ = 1, and the identity tanhh(k+1)=m(k+1), we obtain
pfree(Sε,kc)2kinfλ0expNλεk+Nλ22=2keNε2/(2k2).
This concludes (4.4) for c = ck = 1/(2k2), C = Ck = 2k.□

We note that the constants c, C > 0 in (4.4) are independent of the realization of the disorder {gij}. Thus, a.s. in the disorder (so that ϕ(s), s = 1, …, k, and, hence, ḡ(k+1) and pfree are well-defined), Sɛ,k ≠ ∅ for N large enough.

The next lemma determines the first conditional moment of the reduced partition function ZN(k+1)(Sε,k) and is valid in the full high temperature regime (1.3).

Lemma 4.2.
Letɛ > 0,kN, and letSɛ,k,ZN(k+1)(Sε,k), andpfreebe as in(4.1)(4.3), respectively. Assume that (β, h) satisfy the AT condition(1.3). Then,
limε0limklim supNE1NlogEkZN(k+1)(Sε,k)β24(1q)2=0.
(4.5)

Proof.
By Proposition 3.2, we have that
EkZN(k+1)(Sε,k)=σSε,kpfree(σ)expβ2N24Ekσ,g(k+1)σ2=σSε,kpfree(σ)expβ2N41s=1kσ,ϕ(s)22.
Centering around m(k+1) yields
1s=1kσ,ϕ(s)2=1s=1km(k+1),ϕ(s)22s=1kσm(k+1),ϕ(s)m(k+1),ϕ(s)s=1kσm(k+1),ϕ(s)2
so that
supσSε,k1s=1kσ,ϕ(s)21s=1km(k+1),ϕ(s)2CsupσSε,ks=1k|σm(k+1),ϕ(s)|Cε
for some C > 0 independent of N and k. This implies with Lemma 4.1 that
1NlogEkZN(k+1)(Sε,k)β241s=1km(k+1),ϕ(s)22Cβ2ε+CN|log(1CecNε2)|.
Moreover, by Proposition 3.3, we have that limNs=1km(k+1),ϕ(s)2=Γk2 in Lp(dP) for any p ∈ [1; ∞) so that
lim supNE1NlogEkZN(k+1)(Sε,k)β241Γk22Cβ2ε.
Finally, since limkΓk2=q under the AT condition (1.3), by Lemma 3.1, we let N → ∞, then k → ∞, and then ɛ → 0, which implies that
limε0limklim supNE1NlogEkZN(k+1)(Sε,k)β24(1q)2=0.

The following lemma computes the second conditional moment of ZN(k+1)(Sε,k) under the stronger high temperature condition (1.5).

Lemma 4.3.
Letɛ > 0,kN, and letSɛ,k,ZN(k+1)(Sε,k), andpfreebe as in(4.1)(4.3), respectively. Assume that (β, h) satisfy condition(1.5). Then,
limε0limklim supNE1NlogEkZN(k+1)(Sε,k)2β22(1q)2=0.
(4.6)

Proof.
Proceeding as in the previous proposition, we compute
EkZN(k+1)(Sε,k)2=σ,τSε,kpfree(σ)pfree(τ)expβ2N24Ekσ,g(k+1)σ+τ,g(k+1)τ2=σ,τSε,kpfree(σ)pfree(τ)expβ2N41s=1kσ,ϕ(s)22+β2N41s=1kτ,ϕ(s)22×expβ2N2σ,τs=1kσ,ϕ(s)ϕ(s)τ2=σ,τSε,kpfree(σ)pfree(τ)expβ2N2σ,Q(k)τ2×expβ2N21s=1km(k+1),ϕ(s)22+NO(β2ε).
Arguing as in the previous lemma, we, therefore, see that it is enough to show that
E1Nlogσ,τSε,kpfree(σ)pfree(τ)expβ2N2σ,Q(k)τ2
vanishes when N → ∞. To this end, recall that by the definition of Sɛ,k, we have that
supτSε,kQ(k)τ21m(k+1),P(k)m(k+1)Cε
(4.7)
for some C > 0 independent of N and k. For fixed τSɛ,k, we, then, have
pfree(Sε,k)σSε,kpfree(σ)expβ2N2σ,Q(k)τ2pfree(Sε,k)+01dtNβ2teN2β2t2pfreeσ{1,1}N:|σ,Q(k)τ|>t.
Setting λ = t/(1 − q) and using that log cosh(x + y) ≤ log cosh(x) + y tanh(x) + y2/2 for x,yR, we can estimate the tail probability in the integral by
pfreeσ{1,1}N:|σ,Q(k)τ|>texpNλt+Nλm(k+1),Q(k)τ+Nλ22Q(k)τ2+expNλtNλm(k+1),Q(k)τ+Nλ22Q(k)τ22expNt21q+Nt22Q(k)τ2(1q)2expNt1qQ(k)m(k+1)
so that
pfree(Sε,k)σSε,kpfree(σ)expβ2N2σ,Q(k)τ2pfree(Sε,k)+201dtNβ2teNt22(1q)β2(1q)2+Q(k)τ2(1q)eNt1qQ(k)m(k+1).
In particular, by (4.7) and because
limNm(k+1),P(k)m(k+1)=Γk2,limNQ(k)m(k+1)2=qΓk2,
in Lp(dP) for p ∈ [1; ∞), we obtain under condition (1.5), i.e., β2(1 − q) ≤ 1, that
limε0limklim supNE1Nlogσ,τSε,kpfree(σ)pfree(τ)expβ2N2σ,Q(k)τ2=0,
which implies (4.6).□

Remark.

The key difficulty in the Proof of Lemma 4.3 is to obtain a strong concentration bound on the overlap (σ, τ) ↦ ⟨σ, τ⟩ under the product measure pfree2, restricted to Sɛ,k × Sɛ,k ⊂ {−1,1}N × {−1,1}N. In our proof, the condition β2(1 − q) ≤ 1 emerges due to the restrictions on one of the spin variables, say, τSɛ,k. In principle, one may be able to find the optimal temperature condition by taking into account the restrictions on the other spin variable and using standard large deviation variational estimates, but an exact solution seems difficult. We hope to get back to this point in future work.

In this section, we prove Theorem 1.1 based on Lemmas 4.2 and 4.3. Before we start, let us first re-center the Hamiltonian HN appropriately, as outlined in the Introduction.

Using the notation of Sec. III, we have that

HN(σ)N=β2σ,ḡσ+h,σ=β2σ,ḡ(k+1)σ+β2s=1kσ,ρ̄(s)σ+h,σ.

In contrast to Ref. 16, instead of centering the spins σ around m(k+1), we center the spins in σ,ρ̄(s)σ around γsϕ(s) in order to produce the right cavity field h(k+1). Note that the remaining term σ,ḡ(k+1)σ contains automatically centered spins around s=1kγsϕ(s) (which approximately equals m(k+1)), as ḡ(k+1)ϕ(s)=0 for s < k + 1. We, thus, write

σ,ρ̄(s)σ=2γsσ,ρ̄(s)ϕ(s)+σγsϕ(s),ρ̄(s)(σγsϕ(s))γs2ϕ(s),ρ̄(s)ϕ(s)=2γsσ,ζ(s)+σ̂(s),ρ̄(s)σ̂(s)γs2ϕ(s),ζ(s),

which follows from ρ̄(s)ϕ(s)=ζ(s) and where we set σ̂(s)=σγsϕ(s). Hence,

HN(σ)N=β2σ,ḡ(k+1)σ+h(k+1),σ+β2s=1kσ̂(s),ρ̄(s)σ̂(s)β2s=1kγs2ϕ(s),ζ(s)+βγkqΓk12σ,ζ(k).

Since an exact evaluation of the free energy seems rather involved, let us note here that for configurations σSɛ,k as defined in (4.1), instead, we have approximately HN(σ)/Nβ2σ,ḡ(k+1)σ+h(k+1),σ. Indeed, we find that

σ̂(s),ρ̄(s)σ̂(s)=2σγsϕ(s),ζ(s)(ϕ(s),σγs)ϕ(s),ζ(s)(ϕ(s),σγs)2

with γs ≃ ⟨m(k+1), ϕ(s)⟩. Similarly, recall that ϕ(s),ζ(s)N(0,2/N) for each s and

γkqΓk2supσ{1,1}Nσ,ζ(k)γk+qΓk2ζ(k)

with limk|qΓk2|=limkγk=0 under (1.3) by Lemmas 3.1 and 3.4.

Thus, we obtain the simple lower bound

1NlogZN=1Nlogσ{1,1}NeHN(σ)=log2+1Ni=1Nlogcosh(hi(k+1))+1Nlogσ{1,1}Npfree(σ)eHN(σ)Nh(k+1),σlog2+1Ni=1Nlogcosh(hi(k+1))+1NlogσSε,kpfree(σ)eHN(σ)Nh(k+1),σ

so that

1NlogZNlog2+1Ni=1Nlogcosh(hi(k+1))+1NlogZN(k+1)(Sε,k)Cβεks=1kζ(s)Cβs=1kζ(s)ϕ(s),m(k+1)γsβ2s=1kγs2ϕ(s),ζ(s)βγkζ(k)βqΓk2ζ(k).
(5.1)

We have now all necessary preparations for the Proof of Theorem 1.1.

Proof of Theorem 1.1.

Up to minor modifications, we follow Ref. 16, Sec. 3 and we also abbreviate RS(β,h)=log2+Elogcosh(βqZ+h)+β2(1q)2/4.

By the Paley–Zygmund inequality, we have that
PkZN(k+1)(Sε,k)EkZN(k+1)(Sε,k)/2EkZN(k+1)(Sε,k)24Ek(ZN(k+1)(Sε,k))2.
Given δ1 > 0, Lemmas 4.2 and 4.3 imply that
P2NlogEkZN(k+1)(Sε,k)1Nlog4EkZN2(Sε,k)δ112
if we choose ɛ > 0 sufficiently small, k sufficiently large, and NN1(ε,k)N sufficiently large. This also implies that
PPk1NlogZN(k+1)(Sε,k)1NlogEkZN(k+1)(Sε,k)log2Neδ1N12.
On the other hand, applying Lemma 3.4 (3), we note that
limNP1ks=1kζ(s)>2limNs=1kPζ(s)>2=0
and that
limNPζ(k+1)>2=0.
Moreover, Lemma 3.4 and the fact that ⟨ϕ(s), m(k+1)⟩ ≃ γs by Proposition 3.3 imply that
limNCβs=1kζ(s)ϕ(s),m(k+1)γs+β2s=1kγs2ϕ(s),ζ(s)=0
and
limN1Ni=1Nlogcoshhi(k+1)=Elogcosh(βqZ+h)
in probability. Now, the lower bound (5.1) implies that
PPk1NlogZNlog2+1Ni=1Nlogcosh(βqZ+h)+1NlogEkZN(k+1)(Sε,k)log2NEε,keδ1N12,
where we defined the error Eε,k by
Eε,k=Cβεks=1kζ(s)+Cβs=1kζ(s)ϕ(s),m(k+1)γs+β2s=1kγs2ϕ(s),ζ(s)+βγkζ(k)+βqΓk2ζ(k).
Given δ2 > 0, we may choose ɛ > 0 sufficiently small, kN sufficiently large, and NN2(k,ε)N sufficiently large such that
P|Eε,k|δ24,1Ni=1Nlogcosh(hi(k+1))Elogcosh(βqZ+h)δ2478
and, by Lemma 4.2, also such that
P1NlogEkZN(k+1)(Sε,k)β24(1q)2δ2478.
Combining the above observations, we find that
PPk1NlogZNRS(β,h)δ2eδ1N14
for all N ≥ max(N1(ɛ, k), N2(ɛ, k), 4 log 2/δ2). This implies that
P1NlogZNRS(β,h)δ214eδ1N.
By Gaussian concentration of the free energy, i.e.,
P1NlogZN1NElogZNδ31eNδ32/β2,
we may choose δ1<δ32/(2β2) to conclude for large enough N that
E1NlogZNRS(β,h)δ2δ3.
Since δ2, δ3 > 0 were arbitrary, this shows that the right-hand side of (1.6) is a lower bound to limNE1NlogZN, and by Remark (2), this proves Theorem 1.1.□

The work of H.-T.Y. was partially supported by the NSF under Grant No. DMS-1855509 and a Simons Investigator Award. The work of C.B. was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – GZ 2047/1, Projekt-ID 390685813

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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