Chapter 2

What this is all about: the REM

Consider the space Σ_{N} = {−1, 1}^{N}. A point σ = (σ_{1}, …, σ_{N}) in Σ_{N} will be called a configuration. It physically represents the position (= configuration) of N “spins” σ_{i} = ±1. We will have a collection of r.v. H_{N}(σ), and we will be interested in the smallest, or the few smallest, of the values H_{N}(σ) as σ ranges over all possible configurations. The physical idea is that H_{N}(σ) represents the “energy” of the configuration σ. The function H_{N} is called the Hamiltonian. The energies H_{N}(σ) are random due to some kind of “disorder” that we try to understand. To study the small values of H_{N}(σ) we will weight them using the Boltzmann factors exp(−βH_{N}(σ)), where β ≥ 0 is a parameter. We will then consider Gibbs’ (probability) measure G_{N} on Σ_{N}, that will be given by

G_{N}(σ) = \frac{1}{Z_{N}} exp(−βH_{N}(σ)), (2.1)

where Z_{N} is the normalization factor, called the partition function,

Z_{N} = Z_{N}(β) = ∑ ; σ exp(−βH_{N}(σ)), (2.2)

where the sum is over all configurations σ. The physical idea is that Gibbs’ measure quantifies the thermal fluctuations of the N-spin system with energy levels H_{N}(σ) when it is in thermal equilibrium with a heat bath at temperature T = 1/β. That is, G_{N}(σ) is the probability to find the system in configuration σ after it has been left undisturbed a long time. The − sign is to respect the conventions of physics, where energies are minimized, not maximized.

Gibbs’ measure is a basic concern of statistical mechanics. The reader however will know very soon all the statistical mechanics he will need for these notes. The reason is that disorder (= randomness) is the overwhelming feature here, and that the vast body of knowledge of standard “non disordered” statistical mechanics seems to be largely irrelevant.

Gibbs’ measure is a probability measure. It is itself random. Thus, we have two levels of randomness, and this topic is in some sense a probabilist’s paradise.

Given a (typical) occurrence of the disorder, we will try to understand the structure of G_{N}. The disorder does not evolve with the thermal fluctuations. It is “frozen” or “quenched” as the physicists say. The word “glass” conveys that idea of “frozen disorder”.

The subtilty of definition (2.1) is not apparent at first sight. The normalization will be chosen so that maxσ(−H_{N}(σ)) is of order N so the sum (2.2) is the sum of 2^{N} terms, some of which small, others of exponential order, and it is not clear a priori from which of these comes the main contribution. Computation of Z_{N} is a major objective, and this computation is essentially equivalent to understanding G_{N}, because the derivatives of log Z_{N} with respect to the various parameters we will consider are expressed as integrals with respect to G_{N}. For example

(2.3)  \frac{d}{dβ} log Z_{N}(β) = \frac{1}{Z_{N}(β)} ∑ ; σ −H_{N}(σ) exp(−βH_{N}(σ))
= ⟨−H_{N}(σ)⟩.

In this formula, as well as in the rest of the notes, ⟨ · ⟩ denotes average with respect to Gibbs’ measure, i.e.

⟨f(σ)⟩ = \frac{1}{Z_{N}} ∑ ; σ f(σ) exp(−βH_{N}(σ)). (2.4)

Throughout the notes, we will assume that

(2.5) The family (H_{N}(σ)) is a jointly Gaussian family of r.v.

This family will not always be centered. We will also assume

∀σ, E(H_{N}^{2}(σ)) − (E H_{N}(σ))^{2} ≤ \frac{N}{2} . (2.6)

It should be apparent that E will denote expectation in the variables H_{N}(σ). The corresponding probability will be denoted by P.

It is of course impossible to even think the word Gaussian without immediately mentioning the most important property of Gaussian processes, that is concentration of measure. We will use the following convenient formulation ([I-S-T]).

Proposition 2.1. Consider a function F on ℝ^{M}, and assume that its Lipschitz constant is at most A, i.e.

x, y ∈ ℝ^{M} ⟹ |F (x) − F (y)| ≤ A∥x − y∥ (2.7)

where ∥x∥ is the Euclidean norm of x. Then if g = (g_{1}, …, g_{M}) where the r.v. g_{i} are i.i.d. N (0, 1) we have, for each u > 0,

P(|F (g) − E F (g)| ≥ u) ≤ exp(−\frac{u^{2}}{2A^{2}} ). (2.8)

A statement of similar strength (with the loss of a factor 2 in the exponent, loss that is irrelevant here) will be proved at the end of Chapter 6.

Throughout the paper, we will write

p_{N} = p_{N}(β) = \frac{1}{N} E log Z_{N}(β). (2.9)

Thus p_{N} is the “expected density of the logarithm of the partition function”. It would be interesting to find a nice name for p_{N}. We find it safer to avoid taking unnecessary risks here. We will call this quantity “p_{N}”. The quantity p_{N} is closely related to the “free energy” considered by physicists. The free energy is −\frac{1}{β} p_{N}(β). The factor \frac{1}{β} is a nuisance for mathematics. This had led to the name “free energy” for p_{N} in previous papers but this confusing terminology is abandoned here.

It will turn out that p_{N}(β) is of order 1. The following proposition shows that the single number p_{N}(β) is a good way to capture information about the r.v. \frac{1}{N} log Z_{N}(β).

Proposition 2.2. Under (2.5), (2.6) we have for u ≥ 0,

P(|\frac{1}{N} log Z_{N}(β) − p_{N}(β)| ≥ u) ≤ exp(−\frac{Nu^{2}}{β^{2}} ). (2.10)

Proof. It is elementary that for M = 2^{N} we have a representation

H_{N}(σ) = E H_{N}(σ) + a(σ) · g (2.11)

where a(σ) ∈ ℝ^{M}, g is as in Proposition 2.1, · denotes the dot product and ∥a(σ)∥^{2} ≤ N/2. The function

F (x) = \frac{1}{N} log(∑ ; σ exp(−β(E H_{N}(σ) + a(σ) · x)))

satisfies (2.7) with A = \frac{β}{\sqrt{2N}}, because

|a(σ) · x − a(σ) · y| ≤ \sqrt{\frac{N}{2}} ∥x − y∥.

The result follows from (2.8). □

Comment. The reader who does not find (2.11) obvious should be reassured. We do not provide details because in all cases where (2.10) will actually be used, the representation (2.11) will be obvious from the very definition of H_{N}.

Here is another simple observation:

Proposition 2.3. The (random) functions β ↦ \frac{1}{N} log Z_{N}(β) are convex, and so is p_{N}.

Proof. Let f be a positive function on a measure space. Then, the map β ↦ log(∫ f^{β} dμ) is convex by Hölder’s inequality. □

Thus, the random function \frac{1}{N} log Z_{N}(β) is convex and has small fluctuations. The quantity of information that can be extracted from this simple fact is amazing [G-G].

Here is another simple fact.

Proposition 2.4. For each β ≥ 0 we have

p_{N}(β) ≤ \frac{β^{2}}{4} + \frac{1}{N} log ∑ ; σ exp(−β E H_{N}(σ)). (2.12)

Proof. We use Jensen’s inequality

(2.13) E log Z_{N} ≤ log E Z_{N}

and we use (2.6) and the fact that for a (centered) Gaussian r.v. g we have

E e^{g} = e^{E g^{2}/2}. □ (2.14)

There is a crucial point that must be explained here.

Proposition 2.2 shows that p_{N}(β) is the correct way to obtain information about \frac{1}{N} log Z_{N}(β). However, in general E Z_{N} is not the correct information about Z_{N}. This is because a large part of E Z_{N} comes from very rare events. In other words, the mean and the median of Z_{N} are very different. This is true even when measured on a logarithmic scale. In that scale, the median and the average of Z_{N} are fairly represented respectively by

(2.15) p_{N} = \frac{1}{N} E log Z_{N} and \frac{1}{N} log E Z_{N}.

These two numbers will be of order 1, but different. The interesting (and difficult) problem is to understand the median of Z_{N}. The fundamental difference between these numbers (2.15) could be called the lottery phenomenon. This name reflects the situation of a person holding one of (say) 10^{6} tickets of a lottery that offers a single (huge) prize. His expected gain might be sizable; but his typical gain is zero, significantly lower. The notion of “expected gain” implies that the experiment is repeated many times, and is not the way to look at a single lottery drawing.

For simplicity, we now assume

(2.16) E H_{N}(σ) = 0 for each σ,

so that the variables H_{N}(σ) are centered and E H_{N}^{2}(σ) ≤ N/2.

Proposition 2.5. If β ≥ 0, we have

p_{N}(β) ≤ \frac{β^{2}}{4} + log 2. (2.17)

and if β ≥ 2\sqrt{log 2} we have

p_{N}(β) ≤ β\sqrt{log 2}. (2.18)

Proof. We first observe that (2.17) is a consequence of (2.12) under (2.16). Next we observe that

Z_{N} ≥ exp(β {max \atop σ}(−H_{N}(σ)))

and thus

p_{N}(β) ≥ \frac{β}{N} E {max \atop σ}(−H_{N}(σ)).

Combining with (2.17) for β = 2\sqrt{log 2}, we see that

\frac{1}{N} E {max \atop σ}(−H_{N}(σ)) ≤ \sqrt{log 2}. (2.19)

Next, we use (2.3) to get

(2.20)  \frac{d}{dβ} p_{N}(β) = \frac{1}{N} E⟨−H_{N}(σ)⟩

≤ \frac{1}{N} E({max \atop σ}(−H_{N}(σ)))

≤ \sqrt{log 2},

using (2.18). Combining (2.17) and (2.20) proves (2.18). □

The upper bound of Proposition 2.5 is a kind of worst case. For example, if H_{N}(σ) does not depend upon σ, we have p_{N}(β) = log 2 for each N, β. The interesting (and difficult) case will be when there is some (but not too much) correlation between the variables H_{N}(σ). In the rest of the section, we will investigate the case where

(2.21) The r.v. (H_{N}(σ)) are i.i.d. N (0, N/2).

This case is known as Derrida’s random energy model (REM): the energy levels are random independent. It is not surprising that in this case we can get a complete picture of the situation. This is nonetheless a very instructive exercise.

Our first goal is to show that in this case, the upper bounds of Proposition 2.5 are the correct ones.

Proposition 2.6. For the REM we have

(2.22)  lim_{N→∞} p_{N}(β) = \frac{β^{2}}{4} + log 2 if β ≤ 2\sqrt{log 2}
(2.23)  lim_{N→∞} p_{N}(β) = β\sqrt{log 2} if β ≥ 2\sqrt{log 2}.

Proof. Let us first prove (2.23). We simply use that

p_{N}(β) = \frac{1}{N} E log Z_{N}(β)

≥ \frac{1}{N} E log exp(−β {min \atop σ} H_{N}(σ))

= \frac{β}{N} E {max \atop σ}(−H_{N}(σ)).

That is, we take into account only the largest term of the sum (2.2). Thus to prove (2.23) it is enough to show that the estimate (2.19) is essentially correct when the r.v. H_{N}(σ) are independent. (We do not detail this easy point, as we will prove more later.) Another route is that (2.23) follows from (2.22), the upper bound of (2.20), and the convexity of p_{N}.

Let us now turn to the proof of (2.22). To understand what happens, it is useful to note that in the left-hand side of (2.14), most of the contribution occurs for g ≃ E g^{2}. More precisely we have the following

Lemma 2.7. Consider an N (0, τ^{2}) r.v. g. Then, if c < τ^{2},

E e^{g}1_{\{g≤c\}} ≤ \frac{1}{2} exp(c − \frac{c^{2}}{2τ^{2}} ),

while, if c > τ^{2},

E e^{g}1_{\{g≥c\}} ≤ \frac{1}{2} exp(c − \frac{c^{2}}{2τ^{2}} ).

Thus, if a > 0,

E e^{g}1_{\{|g−τ^{2}|≥a\}} ≤ e^{τ^{2}/2} exp(−\frac{a^{2}}{2τ^{2}} ). (2.24)

Proof. For example, if c > τ^{2}

E e^{g}1_{\{g≥c\}} = \frac{1}{τ\sqrt{2π}} ∫_{c}^{∞} e^{t}e^{−t^{2}/(2τ^{2})} dt

= \frac{1}{τ\sqrt{2π}} e^{τ^{2}/2} ∫_{c}^{∞} exp(−\frac{1}{2τ^{2}} (t − τ^{2})^{2}) dt

= e^{τ^{2}/2} ∫_{(c−τ^{2})/τ}^{∞} \frac{1}{\sqrt{2π}} exp(−\frac{t^{2}}{2} ) dt

≤ \frac{1}{2}e^{τ^{2}/2} exp(−\frac{(c − τ^{2})^{2}}{2τ^{2}} )

= \frac{1}{2} exp(c − \frac{c^{2}}{2τ^{2}} ). □

The meaning of (2.22) is that (on a certain logarithmic scale), most of the time

∑ ; σ exp(−βH_{N}(σ)) ≃ 2^{N} exp \frac{Nβ^{2}}{4} (2.25)

and Lemma 2.7 shows that about the only way this can happen is that there are enough values of σ for which −βH_{N}(σ) ≥ \frac{Nβ^{2}}{2} . This is because if a > 0 the expected contribution of the terms for which −βH_{N}(σ) ≤ \frac{Nβ^{2}}{2} − aN is hopelessly smaller than the right hand side of (2.25). This suggests use of the bound

Z_{N}(β) ≥ exp \frac{Nβ^{2}}{2} card\{σ | −H_{N}(σ) ≥ \frac{Nβ}{2} \} (2.26)

so that

(2.27) \frac{1}{N} log Z_{N}(β) ≥ \frac{β^{2}}{2} + \frac{1}{N} log card\{σ | −H_{N}(σ) ≥ \frac{Nβ}{2} \}.

We now turn to the task of bounding below the card involved in (2.26). Essentially the only known method to do that is called (now) the second moment method, and relies upon the following principle.

Proposition 2.8. Consider a r.v. Y ≥ 0. Then

P(Y ≥ \frac{1}{2} E Y ) ≥ \frac{1}{4} \frac{(E Y)^{2}}{E Y^{2}} . (2.28)

This was apparently used first in the work of Paley and Zygmund on trigonometric series.

Proof. If A = \{Y ≥ \frac{1}{2} E Y \} then

E Y ≤ E(Y 1_{A^{c}}) + E(Y 1_{A}) ≤ \frac{1}{2} E Y + E(Y 1_{A}).

Thus

\frac{E Y}{2} ≤ E(Y 1_{A}) ≤ (E Y^{2})^{1/2} P(A)^{1/2}

using Cauchy-Schwarz. The result follows. □ Proof of (2.22). We will use (2.28) for

Y = card\{σ | −H_{N}(σ) ≥ \frac{Nβ}{2} \} = ∑ ; σ 1_{\{−H_{N} (σ)≥ \frac{Nβ}{2} \}} (2.29)

and thus E Y = 2^{N}Φ(β\sqrt{N/2} ), (2.30) where, if g is N (0, 1),

Φ(t) = P(g ≥ t) = \frac{1}{\sqrt{2π}} ∫_{t}^{∞} exp(−\frac{u^{2}}{2} ) du. (2.31)

Now, from (2.29)

Y^{2} = ∑_{σ,σ′} 1_{\{−H_{N} (σ)≥ \frac{Nβ}{2} , −H_{N} (σ′)≥ \frac{Nβ}{2} \}}

so that

E Y^{2} = ∑_{σ,σ′} A(σ, σ′), (2.32)

where

A(σ, σ′) = P(\{−H_{N}(σ) ≥ \frac{Nβ}{2} , −H_{N}(σ′) ≥ \frac{Nβ}{2} \}).

If σ′ = σ, we have

A(σ, σ) = Φ(β\sqrt{N/2} ).

If σ ≠ σ′, by independence, we have

A(σ, σ′) = Φ(β\sqrt{N/2} )^{2}.

Combining these estimates we get from (2.32) that E Y^{2} ≤ 2^{N}Φ(β\sqrt{N/2}) + 2^{2N}Φ(β\sqrt{N/2})^{2} = E Y + (E Y)^{2}. (2.33) We now use the well known fact that for t > 0,

\frac{1}{L(1 + t)} exp(−\frac{t^{2}}{2} ) ≤ Φ(t) ≤ \frac{1}{2} exp(−\frac{t^{2}}{2} ). (2.34)

Here, as well as in the rest of the paper, L denotes a positive number, not necessarily (and even rarely) the same at each occurrence. Thus, given β < 2\sqrt{log 2}, for N large we have

E Y ≥ \frac{2^{N}}{L\sqrt{N}} e^{−β^{2}N/4} ≥ 1 (2.35)

and (2.28), (2.33) show that

P(Y ≥ \frac{2^{N}}{2L\sqrt{N}} e^{−β^{2}N/4}) ≥ \frac{1}{8}.

Combining with (2.27) we get

P(\frac{1}{N} log Z_{N}(β) ≥ \frac{β^{2}}{4} + log 2 − \frac{1}{N} log(LN)) ≥ \frac{1}{8}.

To prove (2.21), we then observe that by (2.10), we have

P(\frac{1}{N} log Z_{N}(β) ≤ p_{N}(β) + \frac{2β}{\sqrt{N}} ) ≥ 1 − e^{−4} > \frac{7}{8},

and thus

p_{N}(β) ≥ \frac{β^{2}}{4} + log 2 − \frac{2β}{\sqrt{N}} − \frac{1}{N} log(LN).  □

Comments. 1. We have not tried to give the simplest proof or the sharpest result, but we have tried to prepare for future results. Rather than using Proposition 2.8, one can use that E(Y − E Y)^{2} ≤ E Y ≪ (E Y)^{2} to see that P(Y ≤ \frac{E Y}{2} ) is in fact very small.

2. The reader might wonder why we did not use (2.28) for Y = Z_{N}. This is because it is untrue that E Z_{N}^{2} and (E Z_{N})^{2} are of the same order of magnitude if β > \sqrt{2 log 2}. To make (2.28) work directly, one has to use a truncation argument, that our approach was designed to bypass.

3. This scheme of proof would work even if we only knew that E Y^{2} ≤ KN^{2}(E Y)^{2}. This remark will be used in Chapter 4 (see (4.13)).

Having computed p_{N} for Derrida’s REM, we turn to the study of Gibbs’ measure, and we first consider the case β < 2\sqrt{log 2}. Two obvious ways of looking at Gibbs’ measure are respectively the “local way” involving only finitely many coordinates and the “global way” involving all coordinates. These yield very different results. We first consider the “local” point of view. Given a subset I of {1, …, N}, we denote

by G_{I} (= G_{N,I}(β)) the projection of Gibbs’ measure on {−1, 1}^{I};

by μ_{I} the uniform probability on {−1, 1}^{I};
by |G_{I} − μ_{I}| the total variation distance between G_{I} and μ_{I}.

Proposition 2.9. If β < 2\sqrt{log 2} and δ < 1−β^{2}/(4 log 2), there is a number K depending upon β, δ only such that

(2.36) card I ≤ δN ⟹ E |G_{I} − μ_{I}| ≤ exp(−\frac{N}{K} ).

Throughout these notes K will denote a quantity independent of N, but that might depend upon other parameters. This quantity need not be the same at each occurrence. (We recall that L, on the other hand, denotes a universal constant).

Before proving the proposition, we show that the condition

δ < 1 − β^{2}/(4 log 2)

is not absurd. In fact, (2.22) and (2.24) imply that if γ < β/2 then (with probability close to one)

G_{N}(\{σ | −H_{N}(σ) ≤ Nγ\}) ≤ exp(−\frac{N}{K} ),

that is, G_{N} is essentially supported by

A = {σ | −H_{N}(σ) ≥ Nγ} (2.37)

which has the property that

(2.38) E card A ≤ 2^{N} exp(−Nγ^{2}).

Thus, if card I > δ′N, where δ′ > 1 − β^{2}/(4 log 2), we can find γ < β/2 such that

2^{card I} ≫ E card A

and there are not enough points in A to control all the atoms of {−1, 1}^{I}, so (2.36) must fail.

Proof of Proposition 2.9. Of course this can’t be hard, but there is a tiny difficulty due to the fact that second moment computations require truncation (see comment 2 above). We set M = card I. The inverse images of the points of {−1, 1}^{I} under the projection from {−1, 1}^{N} to {−1, 1}^{I} form a family B of subsets of {−1, 1}^{N}. It has 2^{M} elements, each of which contains 2^{N−M} points. We have to show that

E ∑_{B∈B}|2^{−M} − \frac{S(B)}{Z_{N}} | ≤ exp(−\frac{N}{K} ) (2.39)

where S(B) = ∑_{σ∈B} exp(−βH_{N}(σ)).

Consider a number γ with γ < β^{2}/4, that will be determined later. It follows from (2.22), (2.10) that

P(Z_{N} ≤ 2^{N}e^{Nγ}) ≤ exp(−\frac{N}{K} ),

so that to prove (2.39) it is enough to prove that

E ∑_{B∈B}|2^{−M}Z_{N} − S(B)| ≤ 2^{N} exp N(γ − \frac{1}{K} ). (2.40)

Since Z_{N} = ∑_{B∈B} S(B), it suffices to show that for B, B′ ∈ B,

E|S(B) − S(B′)| ≤ 2^{N−M} exp N(γ − \frac{1}{K} ). (2.41)

This is an exercise. It requires truncation. Consider γ′ > β^{2}/2, set

S_{1}(B) = ∑_{σ∈B} exp(−βH_{N}(σ))1_{{−βH_{N} (σ)≥Nγ′}}

and S_{2}(B) = S(B) − S_{1}(B). It follows from Lemma 2.7 that

E S_{1}(B) ≤ 2^{N−M} exp N(γ′ − \frac{γ′^{2}}{β^{2}} ).

Thus, if we assume

γ′ − \frac{γ′^{2}}{β^{2}} < γ, (2.42)

to prove (2.41) it suffices to prove that

E|S_{2}(B) − S_{2}(B′)| ≤ 2^{N−M} exp N(γ − \frac{1}{K} ), (2.43)

or even

E(S_{2}(B))^{2} − (E S_{2}(B))^{2} ≤ (2^{N−M} exp N(γ − \frac{1}{K} ))^{2}. (2.44)

Using independence, for any σ,

E(S_{2}(B))^{2} − (E S_{2}(B))^{2} ≤ 2^{N−M} E exp(−2βH_{N}(σ))1_{{−βH_{N} (σ)≤Nγ′}}

≤ 2^{N−M} exp N(2γ′ − \frac{γ′^{2}}{β^{2}} ),

using Lemma 2.7 with τ^{2} = 2Nβ^{2}, c = 2Nγ′. Thus, to obtain (2.44), it suffices to have that (since M/N = δ)

2γ′ − \frac{γ′^{2}}{β^{2}} < (1 − δ) log 2 + 2γ.

If (1 − δ) log 2 > β^{2}/4, we can find γ < β^{2}/4 and γ′ > β^{2}/2 that satisfy this and (2.42). □

A more tricky question would be to find the largest value of δ such that

E sup_{card I≤δ}|G_{I} − μ_{I}| ≪ 1.

After having looked at G_{N} “locally”, what can we say “globally”? It follows from (2.37) and (2.38) that G_{N} is very far from the uniform measure on Σ_{N}. The reader has certainly already understood that the whole point of studying H_{N} is that it has few large values. These have a dominant influence on Gibbs’ measure. The “peaks” of H_{N} reflect upon the structure of G_{N} at any temperature. We need however to look at the global structure of G_{N} to see this for small β (i.e. β < 2\sqrt{log 2}). This point of view seems to have no motivations from the physical content of the theory, but possibly will raise interesting mathematical problems. It will be discussed again in Chapter 3.

We now turn to the study of Gibbs’ measure at β > 2\sqrt{log 2}. In that case, the most striking feature is that “Gibbs’ measure is supported by a sequence of configurations” in the sense that given ε > 0, one can find k independent of N such that

P (max{G(B) | card B ≤ k} ≤ 1 − ε) ≤ ε. (2.45)

We start with the observation that

P (−H_{N}(σ) ≥ u) = \frac{1}{\sqrt{πN}} ∫_{u}^{∞} exp(−\frac{t^{2}}{N} ) dt, (2.46)

so that if we define a_{N} by Na_{N}^{2} = log(2^{N}/\sqrt{N}), we have

P (−H_{N}(σ) ≥ Na_{N} + u) = \frac{1}{\sqrt{πN}} ∫_{u}^{∞} exp(−\frac{(t + Na_{N})^{2}}{N} ) dt

≤ \frac{2^{−N}}{\sqrt{π}} ∫_{u}^{∞} exp(−2ta_{N}) dt

≤ 2^{−N} exp(−2ua_{N}).

Thus, given k, we have

E(∑ exp(−βH_{N}(σ))1_{{−k+Na_{N} ≥−H_{N} (σ)≥−k−1+Na_{N} }})
≤ 4 exp(βNa_{N} − k(β − 2a_{N})),

and, by summation, since β > 2a_{N},

E(∑ exp(−βH_{N}(σ))1_{\{−H_{N} (σ)≤Na_{N} −k\}}) ≤ K exp(−\frac{k}{K} ) exp βNa_{N}.

The details of the rest of the proof of (2.45) will be left to the reader. If

U = max(−H_{N}(σ))

one shows that P (U ≤ Na_{N} − u) goes to zero uniformly in N as u → ∞. One then observes that

∑{G_{N}(σ) | −H_{N}(σ) ≤ Na_{N} − k}
≤ exp(−βU) ∑ exp(−βH_{N}(σ))1_{{−H_{N} (σ)≤Na_{N} −k}}.
□

The “tightness” result (2.45) can be completed by a limit theorem. Given a bounded Borel subset A of ℝ, the number X_{N}(A) of values of σ for which −H_{N}(σ)−Na_{N} belongs to A converges in law to a Poisson r.v. of expectation

\frac{1}{\sqrt{π}} ∫_{A} exp(−2t\sqrt{log 2}) dt,

and, if B is another Borel subset, disjoint of A, X_{N}(B) and X_{N}(A) are asymptotically independent. In other words, the values taken by −H_{N}(σ)− Na_{N}, for σ ∈ Σ_{N} converge (in each interval [u, +∞)) to a Poisson point process of intensity measure dμ = exp(−2t\sqrt{log 2}) dt. The values taken by the weights exp(−βH_{N}(σ) − βNa_{N}) converge (on each interval [u, +∞) where u > 0) to a Poisson point process having as intensity measure the image of μ under the map x ↦ exp(−βx), which is the measure ν_{m} such that

dν_{m}(x) = \frac{1}{β\sqrt{π}} x^{−m−1} dx (2.47)

for m = 2\sqrt{log 2}/β < 1. The important feature is that, since m < 1, we have

∫ x dν_{m}(x) < ∞,

so that if we denote by (u_{α}) the family of numbers created by a realization of the Poisson point process (2.47), we have ∑ u_{β} < ∞ a.s. We can then define the random weights

v_{α} = \frac{u_{α}}{∑ u_{β}} . (2.48)

If we agree to label them in decreasing order, we thus define a probability measure on the space of sequences (v_{α})_{α≥1}, v_{α} ≥ 0 that sum to one. Going back to Gibbs’ measure, the distribution of the numbers G_{N}(σ), when ordered in decreasing order, converges to the distribution of the weights v_{α} (in a sense that the reader will make precise). The distribution of the weights v_{α} is very natural and interesting (see [P-Y]). It plays a fundamental role at low temperature [T7].