Chapter 7

Central limit theorems and the Almeida-Thouless line

In the previous chapter we have seen how to obtain sharp bounds on certain moments when β ≤ β_{0}. But how do we perform exact computation ? For example, we know since Chapter 5 that for β ≤ 1/3

NC_{N} = N E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2}⟩

remains bounded. We surely expect that this quantity has a limit, and we would like to compute it. Using symmetries between the sites, we write as usual

NC_{N} = E⟨σ_{Ñ}σ_{N}^{∗}σ˜ · σ^{∗}⟩,

and changing N into N + 1 and β in β′,

(N + 1)C_{N+1}(β′) = E⟨σ_{Ñ+1}σ_{N+1}^{∗}σ˜ · σ^{∗}⟩′ + E ⟨(σ_{Ñ+1}σ_{N+1}^{∗})^{2}⟩′. (7.1)

It should be obvious that the last term has a limit, as we will detail later. In view of what we did in Chapter 5, or even in Chapter 6, it is not sufficient to give a bound for the middle term of (7.1). We need an exact expression, or at least an expression with error o(1). We will as usual use (5.22), so we need to evaluate

E \frac{1}{Z^{4}} ⟨σ˜ · σ^{∗} Av ε ε̃^{∗}E_{4}⟩. (7.2)

The closest we can do for the computation of such an expression is in Proposition 6.6. But certainly the proposition (or its obvious extensions to more replicas) is not accurate enough. Since f = σ ˜ · σ^{∗} satisfies ⟨f⟩ = 0, it would suggest 0 as approximation of (7.2), which is not really satisfactory (the error term in (6.19) is of course of order 1). Our first task will then be to obtain a result in the spirit of (6.19), but with higher accuracy. It will be a “second order expansion” rather than a “first order expansion”. The reader who has not followed in detail the computations of the second half of Chapter 6 should at this point skip directly to Corollary 7.4 below. This is the key result that allows to prove Proposition 7.5. This result in turn plays an important role in our discussion of the A-T line, the conjectured separation between high and low temperature behavior. We could perform the main computation in the case of Corollary 7.4 only. Rather, we have decided to give a more general result. There are several good reasons to do this. First, the extra complication is minimal, and purely algebraic. Second, this general result opens the door to interesting computations (Proposition 7.7) which show how far the high temperature phase of the SK model is from being trivial despite what the physicists say.

Our general result will involve s replicas and a given subset J ⊂ {1, …, s} of these replicas. We set n = card J. For l_{1} < l_{2} ≤ s, we set

n(J, l_{1}, l_{2}) = \begin{cases}n + 2 & if l_{1}, l_{2} ∉ J \\ n & if l_{1} ∉ J, l_{2} ∈ J or l_{1} ∈ J, l_{2} ∉ J \\ n − 2 & if l_{1}, l_{2} ∈ J. \end{cases}

If ε_{l} ∈ {−1, 1}, n(J, l_{1}, l_{2}) is the number of terms that actually occur in the product

ε_{l_{1}}ε_{l_{2}} ∏_{l∈J} ε_{l}

after cancelation (ε_{l}^{2} = 1). In a similar manner, we define

n(J, l′) = \begin{cases}n + 1 & if l′ ∉ J \\ n − 1 & if l′ ∈ J. \end{cases}

We recall our usual notation

E_{s} = exp ∑_{l≤s} ε_{l}(\frac{β}{\sqrt{N}} g · σ^{l} + h)

Z = ⟨Av exp ε(\frac{β}{\sqrt{N}} g · σ + h)⟩
\overline{q} = ⟨R(σ, σ′)⟩,

and we recall that q is the solution of (5.4).

To lighten the notation, we write

Y = βg\sqrt{q} + h (7.3) where g is N(0, 1).

Proposition 7.1. Given s, β_{1}, h_{1}, there is a number K = K(s, β_{1}, h_{1}) such

that if β ≤ β_{1}, h ≤ h_{1}, for each function f on Σ_{N}^{s}, each p, p′ with \frac{1}{p} + \frac{1}{p′} = 1,

each subset J of {1, …, s} of cardinal n we have

(7.4)  E \frac{1}{Z^{s}} ⟨f Av(∏_{l∈J} ε_{l})E_{s}⟩

= E th^{n}(Y) E⟨f⟩

+ β^{2} ∑_{1≤l_{1}<l_{2}≤s} E th^{n(J,l_{1},l_{2})}(Y) E⟨f(\frac{σ^{l_{1}} · σ^{l_{2}}}{N} − q)⟩

− sβ^{2} ∑_{1≤l≤s} E th^{1+n(J,l)}(Y) E⟨f(\frac{σ^{l} · b}{N} − q)⟩

+ \frac{s(s + 1)}{2}β^{2} E th^{n+2}(Y) E(⟨f⟩(\overline{q} − q))

+ R,

where

|R| ≤ K E⟨|f|^{p}⟩^{\frac{1}{p}} E⟨|R(σ, σ′) − q|^{2p′}⟩^{\frac{1}{p′}} . (7.5)

Compared with Proposition 6.6, the gain is that we have a better error term, due to the exponent 2p′ rather than p′. But, rather than approximating the left-hand side of (7.4) only by E(th^{n}(Y)⟨f⟩) we now have three extra terms, each of which is a sum. Probably the reader will find that this is awfully complicated. But the model (rather than the author) should be blamed: when ⟨f⟩ = 0 these extra terms are what matters. What will make (7.4) usable is that we use it for combinations of terms such as the left-hand side of (7.4) with various choices of J. These will be chosen such that there is abundant cancelation among the complicated terms. This will be done in Proposition 7.2 and most notably in Proposition 7.3.

Proof of Proposition 7.1. The proof is a refinement of the proof of Proposition 6.6, so that the reader who has not studied this should go directly to Corollary 7.4. We use (6.30), and the same families u, v as in the proof of Proposition 6.6. Rather than (6.38), however, we use a second order expansion

θ(1) = θ(0) + θ′(0) + R (7.6)

where

|R| ≤ |∫_{0}^{1} θ′′(t)(1 − t) dt| ≤ sup_{t}|θ′′(t)|.

It was shown in (6.37) that as M → ∞ and M^{−1} ∑_{j≤M} δ_{σ(j)} converges to Gibbs’ measure, θ(0) contributes as the first term in the right-hand side of (7.4). We compute the contribution of θ′(0). For each r = 1, …, 4, as M → ∞, we examine the limit of E_{g} F_{l_{1},l_{2}}^{r}(u) as M → ∞. For r = 1, this creates the second term on the right-hand side of (7.4). For r = 2, the term A_{l_{1},l_{2}} creates an average ⟨\frac{σ^{1}·σ^{2}}{N} − q⟩ = \overline{q} − q, so this term contributes as \frac{s(s−1)}{2}β^{2} E th^{n+2}(Y) E(⟨f⟩(\overline{q} − q)). For the same reason, the term for r = 4

contributes as s^{2}β^{2} E th^{n+2}(Y) E(⟨f⟩(\overline{q}−q)), and regrouping these gives the penultimate term of (7.4). The term E_{g} F_{l_{1},l_{2}}^{3}(u) has a limit

β^{2}⟨f(\frac{σ^{l_{1}} · σ^{l_{2}+s}}{N} − q)⟩ E th^{m}(Y),

where m is the number of terms ε_{l} occurring in ε_{J}ε_{l_{1}}ε_{l_{2}+s}, that is (since l_{1} ≤ s) 1 + n(J, l_{1}). Since f depends only upon σ^{1}, …, σ^{s}, we have

⟨f(\frac{σ^{l_{1}} · σ^{l_{2}+s}}{N} − q)⟩ = ⟨f(\frac{σ^{l_{1}} · b}{N} − q)⟩,

so that the terms arising from E_{g} F_{l_{1},l_{2}}^{3}(u) contribute as the third term on the right-hand side of (7.4). Thus, to prove Proposition 7.1, it remains to control the error term, that is to show that E|θ′′(t)| is bounded as in (7.5). What one would like to say is that Lemma 6.9 proves that θ′(t) is the sum of pieces of the same nature as θ(t), except that they involve more replicas (up to 3s replicas in (6.44)) and that one has replaced f by f(\frac{σ^{l′}·σ^{l′′}}{N} − q) for certain values l′ ≠ l′′. Thus, applying Lemma 6.9 once more will express θ′′(t) as a sum of similar terms, but now involving yet more replicas, and functions

f(\frac{σ^{l′} · σ^{l′′}}{N} − q)(\frac{σ^{m′} · σ^{m′′}}{N} − q).

Bounding these terms as in the case of θ′(t), but using Hölder’s inequality with coefficients p, 2p′, 2p′ will then yield the required error term

⟨|f|^{p}⟩^{\frac{1}{p}} ⟨|\frac{σ · σ′}{N} − q|^{2p′}⟩^{\frac{1}{p′}} .

It is however not exactly true that F_{l_{1},l_{2}}^{r}(x) is of the same nature as F (x), because F (x) involves different variables x(σ, l) in each replica while this is not the case e.g. for F_{l_{1},l_{2}}^{4}. Thus, before saying that we can iterate Lemma 6.9, we must investigate how to modify this lemma in a situation where different replicas need not have different variables; that is, we consider different variables y(σ, m), m = 1, … and we take x(σ, l) = y(σ, m(l)) where the map l ↦ m(l) need not be one to one. We then see that (6.39) remains valid, provided that in the first two terms in the right-hand side of (6.39) we restrict the summation to the pairs l_{1} < l_{2} with m(l_{1}) ≠ m(l_{2}). This completes the proof. □

Proposition 7.2. We have

(7.7)  E \frac{1}{Z^{s}} ⟨f Av ε ε̃^{3}E_{s}⟩ = β^{2} E \frac{1}{ch^{2} Y} E⟨f \frac{σ˜ · σ^{3}}{N} ⟩

+ β^{2} ∑_{l≥4} E \frac{th^{2} Y}{ch^{2} Y} E⟨f \frac{σ˜ · σ^{l}}{N} ⟩

− sβ^{2} E \frac{th^{2} Y}{ch^{2} Y} E⟨f \frac{σ˜ · b}{N}⟩ + R,

where R is as in (7.5).

Proof. Let J_{1} = {1, 3}, J_{2} = {2, 3}. From (7.4) we get

(7.8)E \frac{1}{Z^{s}} ⟨f Av ε ε̃^{3}E_{s}⟩

= β^{2} ∑_{1≤l_{1}<l_{2}≤s} E((th Y)^{n(J_{1},l_{1},l_{2})} − (th Y)^{n(J_{2},l_{1},l_{2})}) E⟨f(\frac{σ^{l_{1}} · σ^{l_{2}}}{N} − q)⟩

− sβ^{2} ∑_{1≤l≤s} E((th Y)^{1+n(J_{1},l)} − (th Y)^{1+n(J_{2},l)}) E⟨f(\frac{σ^{l} · b}{N} − q)⟩ + R.

Now n(J_{1}, l_{1}, l_{2}) = n(J_{2}, l_{1}, l_{2}) unless l_{1} ≤ 2 < l_{2}; and n(J_{1}, l) = n(J_{2}, l) unless l = 1, 2. To obtain the result, we regroup the contributions of l_{1} = 1, l_{1} = 2 (treating separately the case l_{2} = 3 or l_{2} > 3) and the contributions of l = 1, 2. We then use that

1 − th^{2} Y = \frac{1}{ch^{2} Y} .  □

Proposition 7.3. We have

E \frac{1}{Z^{s}} ⟨f Av ε ε̃^{∗}E_{s}⟩ = β^{2} E \frac{1}{ch^{4} Y} E⟨f \frac{σ˜ · σ^{∗}}{N} ⟩ + R, (7.9)

where R is as in (7.5).

Proof. Use (7.7) twice on ε ε̃^{3} and ε ε̃^{4}, and subtract. □

Corollary 7.4. Under the conditions of Proposition 7.1, we have

E \frac{1}{Z^{s}} ⟨f Av ε ε̃^{∗}E_{s}⟩ = β^{2} E \frac{1}{ch^{4} Y} E⟨f \frac{σ˜ · σ^{∗}}{N} ⟩ + R, (7.10)

where

R ≤ K \frac{p′}{N} (E⟨|f|^{p}⟩)^{\frac{1}{p}} .

Proof. We have shown in Proposition 6.2 that

E⟨|\frac{σ · σ′}{N} − q|^{2k}⟩ ≤ (\frac{Lk}{N} )^{k}.

If k is the smallest integer such that p′ ≤ k we write

E⟨|\frac{σ · σ′}{N} − q|^{2p′}⟩^{\frac{1}{p′}} ≤ E⟨|\frac{σ · σ′}{N} − q|^{2k}⟩^{\frac{1}{k}} ≤ \frac{Lk}{N}

to control the remainder (7.5). □

We now go back to the study of (7.1), when β ≤ β_{0}, h ≤ 1 (so that we can use Corollary 7.4). We use (7.10) for s = 4, f = σ˜ · σ^{∗}, p = p′ = 2. We know that E⟨f^{2}⟩ ≤ LN by Theorem 6.1, so that (7.1) yields

(N + 1)C_{N+1}(β′) = β^{2} E \frac{1}{ch^{4} Y} NC_{N}(β) + E⟨(σ_{Ñ+1}σ_{N+1}^{∗})^{2}⟩′ + R, (7.11)

where |R| ≤ K/\sqrt{N}. Let us compute the last term. With the notation ε_{l} = σ_{N+1}^{l}, and since ε ^{˜ 2} = 2(1 − ε_{1}ε_{2}), it is

E \frac{1}{Z^{4}} ⟨Av(ε ε̃^{∗})^{2}E_{4}⟩ = 4 E \frac{1}{Z^{4}} ⟨Av(1 − ε_{1}ε_{2} − ε_{3}ε_{4} + ε_{1}ε_{2}ε_{3}ε_{4})E_{4}⟩,

and using (7.4) when f = 1, we see that

E⟨(σ_{Ñ+1}σ_{N+1}^{∗})^{2}⟩′ = 4 E \frac{1}{ch^{4} Y} + R, (7.12)

where |R| ≤ K/\sqrt{N} (in fact, |R| ≤ K/N). Now we see that we are on the right track, but (7.11) alone would require the use of iteration. To avoid this, we will show that (N + 1)C_{N+1}(β′) ≃ NC_{N}(β). We write

NC_{N} = E⟨\frac{1}{N} ∑_{i≤N} σ_{i}˜ σ_{i}^{∗}σ˜ · σ^{∗}⟩ = E⟨\frac{1}{N − 1} ∑_{i≤N−1} σ_{i}˜ σ_{i}^{∗}σ˜ · σ^{∗}⟩, (7.13)

using symmetry between sites. Changing N into N + 1, we get

(7.14) (N + 1)C_{N+1}(β′) = E⟨\frac{1}{N} σ˜ · σ^{∗}(σ˜ · σ^{∗} + σ_{Ñ+1}σ_{N+1}^{∗})⟩′

= N E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2}⟩′ + \frac{1}{N} E⟨σ_{Ñ+1}σ_{N+1}^{∗}σ˜ · σ^{∗}⟩′.

We leave it to the reader to use (7.4), with n = 0, to obtain

|(N + 1)C_{N+1}(β′) − NC_{N}(β)| ≤ KN E⟨|\frac{σ · σ′}{N} − q|^{3}⟩, (7.15)

so that, using Proposition 6.2, we have

|(N + 1)C_{N+1}(β′) − NC_{N}(β)| ≤ \frac{K}{\sqrt{N}} . (7.16)

We combine with (7.11), (7.12), to see that

NC_{N} = β^{2} E \frac{1}{ch^{4} Y} NC_{N} + 4 E \frac{1}{ch^{4} Y} + R, (7.17)

with |R| ≤ K/\sqrt{N}. We have proved the following:

Proposition 7.5. If β ≤ β_{0}, h ≤ 1, we have

lim_{N→∞} N E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2}⟩ = \frac{4A}{1 − β^{2}A}, (7.18)

where

A = E \frac{1}{ch^{4} Y} = E \frac{1}{ch^{4}(βg\sqrt{q} + h)}.

The importance of this result is that it provides an identification of the so-called Almeida-Thouless line. This line is given by the relation

β^{2} E \frac{1}{ch^{4}(βg\sqrt{q} + h)} = 1, (7.19)

where q is “the” root of (5.4). It is certainly not obvious that (5.4) has a unique root; This was recently proved by R. Latala (private communication). Since the left-hand side of (7.18) is a square, (7.18) can hold only if the righthand side is positive, that is if

β^{2} E \frac{1}{ch^{4}(βg\sqrt{q} + h)} < 1. (7.20)

The physicists believe that (5.4) holds under (7.20) (“the high temperature region”), but fails if

β^{2} E \frac{1}{ch^{4}(βg\sqrt{q} + h)} > 1, (7.21)

(“the low temperature region”). It is certain that under (7.21), (7.18) cannot hold, so one of the facts we have used in its derivation must be wrong. Using (7.17), that we read as

NC_{N} = BNC_{N} + 4 E \frac{1}{ch^{4} Y} + R, (7.22)

where B > 1, we see that we can take advantage of this only if R ≪ NC_{N}, or, rather, that (7.22) implies that R is of the same order as NC_{N}. This error term has several sources. One of them is (7.5), where we have used f = σ˜ · σ^{∗}. The natural choice of p = 2 = p′ there gives an error term

\sqrt{C_{N}}(E⟨(\frac{σ · σ′}{N} − q)^{4}⟩)^{\frac{1}{2}} . (7.23)

There is also an error term from (7.15), but we can argue that this is less important. What we would really like to show is that under (7.21) C_{N} cannot be small, because the fact “C_{N} small” is really the main reason behind (5.3). So we try to argue by contradiction. If C_{N} is small (lim C_{N} = 0 as N → ∞) does it follow that the quantities (7.23) are much smaller than NC_{N}? A first difficulty is that a priori it is not clear how to relate

E ⟨(\frac{σ · σ′}{N} − q)^{4}⟩ (7.24)

and

E⟨(\frac{σ˜ · σ^{∗}}{N} )^{4}⟩. (7.25)

(See Lemma 5.3). But even if we know that the quantities (7.24), (7.25) are of the same order, it is not clear why we should have

(7.26) E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2}⟩ small ⟹ E⟨(\frac{σ˜ · σ^{∗}}{N} )^{4}⟩ ≪ E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2}⟩.

In other words, are terms that look second order terms really second order terms? This at first looks like a purely technical point. Rather, as will be explained in detail later in this chapter, this appears to be a question of fundamental importance.

Let us now turn to the discussion of the following important issue: is it true that (5.3) holds under (7.20)? The following is probably more relevant

Problem 7.6. Is it true that in the region (7.20) we have

lim_{N→∞} E⟨(R(σ, σ′) − q)^{2}⟩ = 0?

This problem, to a large extend, was recently solved in [T8]. Nonetheless, it seems worthwhile to discuss in detail the underlying issues. The question of controlling the entire “high temperature” region is of fundamental interest for many models, and the solution given in [T8] is unfortunately very specific to the SK model.

We have found condition (7.20) through the computation (7.18). A first obvious question is: couldn’t we find a more stringent condition through another computation? Apparently this is not the case. It seems desirable that the function Ψ of (5.57) be a contraction near q, and Lemma 5.10 shows that this amounts to

β^{2} E \frac{2 sh^{2}(β\sqrt{q} + h) − 1}{ch^{4}(βg\sqrt{q} + h)} > −1.

The physicists say that this holds under (7.20). They have probably checked this numerically.

To illustrate the fact that not everything is so simple, we now start another interesting computation, that of

lim_{N→∞} N E⟨(\frac{σ˜ · σ^{3}}{N} )^{2}⟩ = lim_{N→∞} ND_{N}. (7.27)

The computation will be valid when β ≤ β_{0}, h ≤ 1. The error terms are controlled as in the computation (7.18), so we will not explain again how this is done each time. It always follows from Proposition 6.2 and Hölder’s inequality. We will denote by R a quantity such that |R| ≤ K/\sqrt{N}. We start as usual with the relation

ND_{N} = E⟨(\frac{1}{N} ∑_{i≤N} σ_{i}˜ σ_{i}^{3})σ˜ · σ^{3}⟩ = E⟨σ_{Ñ}σ_{N}^{3}σ˜ · σ^{3}⟩,

so that

(N + 1)D_{N+1}(β′) = E⟨σ_{Ñ+1}σ_{N+1}^{3}σ˜ · σ^{3}⟩′ + E⟨(σ_{Ñ+1}σ_{N+1}^{3})^{2}⟩′. (7.28)

Now, (σ_{Ñ+1}σ_{N+1}^{3})^{2} = 2(1 − σ_{N+1}^{1}σ_{N+1}^{2}), and as in (7.12) we obtain

E⟨(σ_{Ñ+1}σ_{N+1}^{3})^{2}⟩′ = 2 E \frac{1}{ch^{2} Y} + R. (7.29)

Next, we use Proposition 7.2 to write

(7.30)  E⟨σ_{Ñ+1}σ_{N+1}^{3}σ˜ · σ^{3}⟩′ = E \frac{1}{Z^{3}} ⟨σ˜ · σ^{3} Av ε ε̃^{3}E_{3}⟩

= β^{2} E \frac{1}{ch^{2} Y} ND_{N}

− 3β^{2} E \frac{th^{2} Y}{ch^{2} Y} E⟨σ˜ · σ^{3} \frac{σ˜ · b}{N}⟩ + R.

Now, as in (7.15) we find

(N + 1)D_{N+1}(β′) = ND_{N} + R, (7.31)

and combining with (7.28) we have

(7.32)  ND_{N} = (β^{2} E \frac{1}{ch^{2} Y} )ND_{N}

− \frac{3β^{2}}{N} E \frac{th^{2} Y}{ch^{2} Y} E⟨σ˜ · σ^{3} σ˜ · b⟩ + 2 E \frac{1}{ch^{2} Y} + R.

Now we observe that

⟨(σ˜ · σ^{∗})^{2}⟩ = ⟨(σ˜ · σ^{3} − σ˜ · σ^{4})^{2}⟩
= 2⟨(σ˜ · σ^{3})^{2}⟩ − 2⟨σ˜ · σ^{3} σ˜ · σ^{4}⟩,

so that

E⟨σ˜ · σ^{3}σ˜ · b⟩ = E⟨σ˜ · σ^{3} σ˜ · σ^{4}⟩

= N^{2}D_{N} − \frac{1}{2}N^{2}C_{N},

and combining with (7.32) we have

ND_{N}(1 − β^{2} E(\frac{1 − 2 sh^{2} Y}{ch^{4} Y} )) = \frac{3}{2}β^{2} E \frac{th^{2} Y}{ch^{2} Y} NC_{N} + 2 E \frac{1}{ch^{2} Y} + R. (7.33)

In order to make the point that not everything is very simple, we state the following conclusion, that follows from (7.33) and (7.18).

Proposition 7.7. If β ≤ 1/10 and h ≤ 1, we have

lim_{N→∞} N E⟨(\frac{σ˜ · σ^{3}}{N} )^{2}⟩ = \frac{2B − 6β^{2}A^{2} + 4β^{2}AB}{(1 − β^{2}A)(1 − β^{2}(3A − 2B))} (7.34)

where A is as in Proposition 7.5 and

B = E \frac{1}{ch^{2} Y} .

Challenge problem 7.8. Compute

lim_{N→∞} N E⟨(\frac{σ^{1} · σ^{2} − ⟨σ^{1} · σ^{2}⟩}{N})^{2}⟩.

And what about

lim_{N→∞} N E⟨(\frac{σ^{1} · σ^{2} − E⟨σ^{1} · σ^{2}⟩}{N})^{2}⟩?

Do you find a new constraint other than (7.20)? Hint: [T4], Section 6.

The main difficulty in proving that Problem 7.6 has a positive answer is that it seems very difficult to do any kind of computation about C_{N} unless one also has a control about

E⟨(\frac{σ˜ · σ^{∗}}{N} )^{4}⟩,

or, even better, about

E⟨(R(σ, σ′) − q)^{4}⟩.

In turn, it seems possible to control these only by controlling quantities where 4 is replaced by 6, etc. This was one motivation behind the work of Chapter 6.

Everything would be easy if it were true that (7.26) holds. At first sight the failure of (7.26) looks like a rather remote possibility, one of these pathologies that are “in principle” possible, and that only weird people (such as mathematicians) care about. This is however not the case. The failure of (7.26) is not only possible. It is also natural. It does represent a different type of transition from high to low temperature than that predicted for the SK model. To understand what happens, we will compare the behavior of the SK model and the p-spin interaction model as β increases to the critical value β_{p}, and p is even. We will assume that h = 0, because then all the claims we make can be proved rather than conjectured. We expect that if h is small the situation is similar.

When h = 0 we have b = ⟨σ⟩ = 0 by symmetry. There is no “centering problem” here and the natural quantity to measure the correlation of spins is

E⟨R(σ, σ′)^{2}⟩ = E⟨(\frac{σ · σ′}{N} )^{2}⟩

= \frac{1}{N} + \frac{N − 1}{N} E(⟨σ_{i}σ_{j}⟩^{2}).

Proposition 7.9. If h = 0, β < 1, for the typical value of the disorder, and N large, the map

(σ, σ′) ↦ \frac{σ · σ′}{\sqrt{N}}

resembles a N (0, \frac{1}{1 − β^{2}} ) r.v. More precisely, for each k ≥ 0, the r.v.

⟨(\frac{σ · σ′}{\sqrt{N}} )^{2k}⟩

converges in probability to (1 − β^{2})^{−2k} E g^{2k} where g is N (0, 1).

This is proved in [C-N]. When h ≠ 0, we will prove a similar result (Theorem 7.12 below). The difference is that Theorem 7.12 is proved only for β < 1/10, while Proposition 7.9 holds for β < 1. (See Challenge problem 7.15 to understand better the relationship).

In other words, what Proposition 7.9 says is that N^{−1/2}σ · σ′ is distributed like C(β)g (where g is N (0, 1)), where C(β) → ∞ as β gets close to the critical value β_{c}. What happens for large p is totally different.

Proposition 7.10. For the p-spin interaction model with h = 0, there exist L > 0, p_{0} and x > 0 such that if p ≥ p_{0} and β ≤ 2^{p/2−6}, we have

E(R(σ, σ′)^{2}1_{\{|R(σ,σ′)|≤x\}}) ≤ \frac{L}{N} . (7.35)

If moreover β > 2\sqrt{log 2}, we have

lim inf_{N→∞} E(⟨1_{{|R(σ,σ′)|≥x}}⟩) > 0. (7.36)

Comment. In particular if p is large, there are situations where (7.35) and (7.36) occur simultaneously.

Proposition 7.10 says that if we do not look at the values of R(σ, σ′) that are of order 1 we do not see any increase in the fluctuations of this quantity as β increases beyond the critical value β_{c}. On the other hand, values of R(σ, σ′) of that order exist for β > β_{c}. We will prove Proposition 7.10 in Chapter 8.

Proposition 7.11. For the 4-spin interaction model there exist arbitrarily large values of N for which for certain values of β,

E(⟨R(σ, σ′)^{2}⟩) ≤ \frac{L}{N} ,

but

E(⟨R(σ, σ′)^{2}⟩) ≤ L E(⟨R(σ, σ′)^{4}⟩). (7.37)

This in particular shows that (7.26) cannot follow from general principles.

If we do not insist on p = 4, but allow p to be large, this is a consequence of Proposition 7.10 as we now show. Let us assume x irrational, so that we never have R(σ, σ′) = Nx. Let us consider N and β_{0} with 0 ≤ β_{0} < β_{p} + \frac{1}{2} and

E(⟨1_{\{|R(σ,σ′)|≥x\}}⟩_{β_{0}}) ≥ \frac{1}{N} . (7.38)

Here, the subscript β_{0} indicates of course that Gibbs’ measure is taken at this value of the parameter β.

Since N is fixed, the quantity

E(⟨1_{{|R(σ,σ′)|≥x}}⟩_{β}) (7.39)

is a continuous function of β. It is exponentially small at β = 0, so we can fix β ≤ β_{0} < β_{p} + \frac{1}{2} for which

μ := E(⟨1_{\{|R(σ,σ′)|≥x\}}⟩) = \frac{1}{N} . (7.40)

(As β is now fixed, it is no longer indicated in the notation). We then deduce (7.37) from the fact that

E(⟨R(σ, σ′)^{p}⟩) ≥ x^{p}μ ≥ \frac{x^{p}}{N} ,

while

E(⟨R(σ, σ′)^{2}⟩) ≤ E(⟨R(σ, σ′)^{2}1_{\{|R(σ,σ′)|^{2}≤x\}}⟩) + μ ≤ \frac{L + 1}{N}.  □

It is nonetheless very instructive to give a direct proof of Proposition 7.11. This will also be done in Chapter 8.

The preceding discussion has shown that, as β increase, there are (at least) two very distinct ways in which the structure described in Chapter 6 can deteriorate. Condition (7.21) identifies the value of β where deterioration will occur following the pattern of Proposition 7.5. The challenge of Problem 7.6 is to show that the other type of deterioration, following the pattern of Proposition 7.10 will not occur first.

We now give an example of computation of higher moments.

Theorem 7.12. If β ≤ β_{0}, h ≤ 1, for n = 1, 2 we have

lim_{N→∞} N^{nk} E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2k}⟩^{n} = (\frac{4A}{1 − β^{2}A})^{nk}(E g^{2k})^{n} (7.41)

where g is N (0, 1) and A is as in Proposition 7.5.

This implies that for each k, the r.v.

M_{k} = N^{k}⟨(\frac{σ˜ · σ^{∗}}{N} )^{2k}⟩

satisfies E M_{k}^{2} ≃ (E M_{k})^{2} so that M_{k} converges in probability to E M_{k}. Given k_{0}, for large N and the typical value of the disorder, the moments of order k ≤ k_{0} of the function

(σ^{1}, …, σ^{4}) ↦ \frac{σ˜ · σ^{∗}}{\sqrt{N}}

are approximately those of a N (0, 4A/(1 − β^{2}A)) r.v. Thus, Theorem 7.12 is indeed a central limit theorem.

Proof of Theorem 7.12. For k_{1}, k_{2} ≥ 0, we prove that

(7.42)  lim_{N→∞} N^{k_{1}+k_{2}} E⟨(\frac{σ˜ · σ^{∗}}{N} )^{2k_{1}}⟩⟨(\frac{σ˜ · σ^{∗}}{N} )^{2k_{2}}⟩

= (\frac{4A}{1 − β^{2}A})^{k_{1}+k_{2}} E g^{2k_{1}} E g^{2k_{2}}.

The proof is by induction over k = k_{1} + k_{2}. For k = 1, this is Proposition 7.5. We write

a_{k_{1},k_{2},N} = a_{k_{1},k_{2},N}(β) = E⟨(σ˜ · σ^{∗})^{2k_{1}}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩.

We use the symmetry between sites to write

a_{k_{1}+1,k_{2},N} = N E⟨σ_{Ñ}σ_{N}^{∗}(σ˜ · σ^{∗})^{2k_{1}+1}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩,

and, changing N into N + 1,

a_{k_{1}+1,k_{2},N+1}(β′) = (N + 1) E ⟨σ_{Ñ+1}σ_{N+1}^{∗}(σ˜ · σ^{∗} + σ_{Ñ+1}σ_{N+1}^{∗})^{2k_{1}+1}⟩′

⟨(σ˜ · σ^{∗} + σ_{Ñ+1}σ_{N+1}^{∗})^{2k_{2}}⟩′. □ (7.43)

Lemma 7.13. We have

E⟨|σ˜ · σ^{∗}|^{a}⟩′⟨|σ˜ · σ^{∗}|^{b}⟩′ ≤ KN^{\frac{(a+b)}{2}} .

Proof. We use Cauchy-Schwarz to reduce this to the proof of

E⟨|σ˜ · σ^{∗}|^{2a}⟩′ ≤ KN^{a}.

We use (5.22) and that Z ≥ 1. We first integrate w.r. to g to reduce the proof to the fact that E⟨|σ˜ · σ^{∗}|^{2a}⟩ ≤ KN^{a}, which follows from Proposition 6.2. □

We want to study

lim_{N→∞}(N + 1)^{−k_{1}−k_{2}−1}a_{k_{1}+1,k_{2},N+1}.

We expand the powers in (7.43) and we use that |σ_{Ñ+1}σ_{N+1}^{∗}| ≤ 4 and Lemma 7.13 to see that

a_{k_{1}+1,k_{2},N+1}(β′) (7.44)

= (N + 1) E⟨σ_{Ñ+1}σ_{N+1}^{∗}(σ˜ · σ^{∗})^{2k_{1}+1}⟩′⟨(σ˜ · σ^{∗})^{2k_{2}}⟩′
+ (2k_{1} + 1)(N + 1) E⟨(σ_{Ñ+1}σ_{N+1}^{∗})^{2}(σ˜ · σ^{∗})^{2k_{1}}⟩′⟨(σ˜ · σ^{∗})^{2k_{2}}⟩′
+ 2k_{2}(N + 1) E⟨σ_{Ñ+1}σ_{N+1}^{∗}(σ˜ · σ^{∗})^{2k_{1}+1}⟩′
⟨σ_{Ñ+1}σ_{N+1}^{∗}(σ˜ · σ^{∗})^{2k_{2}−1}⟩′
+ R
= I + II + III + R,

where

|R| ≤ KN^{k_{1}+k_{2}+ \frac{1}{2}} . (7.45)

We will study each of the terms I to III. To reduce the product of brackets to a single bracket we set

U = ((σ^{5} − σ^{6}) · (σ^{7} − σ^{8}))^{2k_{2}},

so that

I = (N + 1) E⟨σ_{Ñ+1}σ_{N+1}^{∗}(σ˜ · σ^{∗})^{2k_{1}+1}U⟩′.

We use the cavity method and Proposition 7.3 to obtain

I = \frac{N + 1}{N}β^{2} E \frac{1}{ch^{4} Y} E⟨(σ˜ · σ^{∗})^{2k_{1}+2}U⟩ + R, (7.46)

where R is as in (7.5), i.e.

|R| ≤ L(N + 1) E(⟨|(σ˜ · σ^{∗})^{2k_{1}+1}U|^{p}⟩)^{1/p} E⟨|\frac{σ · σ′}{N} − q|^{2p′}⟩^{1/p′}. (7.47)

Taking p = p′ = 2, and using Hölder’s inequality, we see that |R| is as in (7.45). It is essential here (as was already essential in Proposition 7.5) to have 2p′ rather than p′ in (7.47).

The term II is less dangerous, because σ˜ ·σ^{∗} occurs with a lower power. It is enough there to use Proposition 6.6 to get

II = (2k_{1} + 1)(N + 1) E \frac{1}{ch^{4} Y} E⟨(σ˜ · σ^{∗})^{2k_{1}}U⟩ + R, (7.48)

where R is as before.

To study the term III, we also use Proposition 6.6. The difference with the term II is that the main contributions cancel out. Writing ε_{l} for σ_{N+1}^{l}, the dependence in the variables ε_{l} is as in

(ε_{1} − ε_{2})(ε_{3} − ε_{4})(ε_{5} − ε_{6})(ε_{7} − ε_{8})

and out of the 16 resulting terms, each involving a product of 4 different ε_{l}, 8 of them have a negative sign.

Now we have obtained

a_{k_{1}+1,k_{2},N+1}(β′) = \frac{N + 1}{N} E \frac{β^{2}}{ch^{4} Y} E⟨(σ˜ · σ^{∗})^{2k_{1}+1}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩

+ (N + 1)(2k_{1} + 1) E \frac{4}{ch^{4} Y} E⟨(σ˜ · σ^{∗})^{2k_{1}+1}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩

+ R.

(7.49)  a_{k_{1}+1,k_{2},N+1}(β′) = β^{2} E \frac{1}{ch^{4} Y} a_{k_{1}+1,k_{2},N}(β)

+ (2k_{1} + 1)N E \frac{4}{ch^{4} Y} a_{k_{1},k_{2},N}(β) + R.

Now we claim that

a_{k_{1}+1,k_{2},N+1}(β′) = a_{k_{1}+1,k_{2},N}(β) + R. (7.50)

Indeed, we write

a_{k_{1}+1,k_{2},N}(β′) = \frac{N}{N − 1} E⟨ ∑_{i≤N−1} σ_{i}˜ σ_{i}^{∗}(σ˜ · σ^{∗})^{2k_{1}+1}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩, (7.51)

we replace N by N + 1; we use Lemma 7.13 to see that only the term

\frac{N + 1}{N} E⟨(σ˜ · σ^{∗})^{2k_{1}+2}⟩⟨(σ˜ · σ^{∗})^{2k_{2}}⟩

matters, and we conclude with Proposition 6.6 again.

Combining (7.49), (7.50), we have reached the conclusion that

(N + 1)^{−k_{1}−k_{2}−1}a_{k_{1}+1,k_{2},N}(1 − β^{2} E \frac{1}{ch^{4} Y} ) (7.52)

= (2k_{1} + 1) E \frac{4}{ch^{4} Y} N^{−k_{1}−k_{2}}a_{k_{1},k_{2},N} + R,

where

|R| ≤ K/\sqrt{N}.

We also observe that by integration by parts,

E(g^{2k_{1}+2}) = E(gg^{2k_{1}+1}) = (2k_{1} + 1) E(g^{2k_{1}}).

Thus (7.52) finishes the induction step. □ Challenge problem 7.14. Prove Proposition 7.9. Hint: Use the results of Chapter 3. Challenge problem 7.15. Compute the following, when β < β_{0}:

lim_{N→∞} N E⟨(\frac{σ^{1} · σ^{2}}{N} − q)^{2}⟩; lim_{N→∞} N E⟨(\frac{σ^{1} · b}{N} − q)^{2}⟩

lim_{N→∞} N E(\overline{q} − q)^{2}.

If you don’t have the stamina to compute these limits, at least prove their existence.