Renaud Leplaideur
email: renaud.leplaideur@univ-brest.fr

F R Éd

Érique Watbled
email: watbled@univ-ubs.fr

EASY ISING OR CURIE-WEISS MODEL EXPLAINED TO ERGODICISTS

Keywords: 2010 Mathematics Subject Classification. 37A35, 37A50, 37A60, 82B20, 82B30, 82C26 thermodynamic formalism, equilibrium states, Curie-Weiss model, Curie-Weiss-Potts model, Gibbs measure, phase transition

We state a dictionary for thermodynamic formalism within the Curie-Weiss model between the Probability-Statistical Mechanics and the Ergodic viewpoints. Limits for probabilistic Gibbs measures are identified as combinations of, say N (β), dynamical conformal measures. Phase transitions are then related to a change in N (β) as β moves.

More surprising we point out that, to be closer to what is done in Statistical Mechanics, ergodicists should study the supremum of the sum of the entropy and the square of the integral of the potential instead of the sum of the entropy and the integral of the potential.

1. Introduction 1.1. Background, main motivations and results. The notion of Gibbs measure comes from Statistical Mechanics. It has been studied a lot from the probabilistic viewpoint (see [START_REF] Georgii | Gibbs measures and phase transitions[END_REF][START_REF] Costeniuc | Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model[END_REF][START_REF] Ellis | Limit theorems for sums of dependent random variables occurring in statistical mechanics[END_REF][START_REF] Ellis | The statistics of Curie-Weiss models[END_REF]). This notion was introduced in Ergodic Theory in the 70's by Sinai, Ruelle and Bowen (see [START_REF] Sinai | Gibbs measures in ergodic theory[END_REF][START_REF] Sinaȋ | Theory of phase transitions: rigorous results[END_REF][START_REF] Ruelle | Thermodynamic formalism[END_REF][START_REF] Ruelle | Statistical mechanics[END_REF][START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]). Since that moment, the thermodynamic formalism became in Dynamical Systems a pure mathematical question and somehow disconnected from the original physical questions.

Since the 00's, ergodicists started to reconnect the questions in Dynamical Systems with the original physical ones. Notions as ground states and phase transitions were re-defined and investigated from the ergodic viewpoint. From that moment, the sharing of the vocabulary started to be a source of confusion, and the main motivation for this paper is to make clear some similitudes and differences between Gibbs measures within the probabilistic or the ergodic viewpoints.

Beyond stating a kind of dictionary between thermodynamic formalisms in Statistical Mechanics and Probability on the one hand, and Ergodic Theory on the other hand, the main byproduct of our work is a very strange discovery within the ergodic viewpoint.

Actually, and using the Ergodic viewpoint and vocabulary, Theorem 2 shows that, in Statistical Mechanics and Probability, one maximizes the entropy plus the square of an integral, whereas in Ergodic Theory one maximizes the entropy plus an integral. This has a lot of nice consequences and further possible research questions. Some of them are discussed later (see Subsubsection 1.2.3).

Our dictionary works as follows. In Statistical Mechanics and in Probability theory, the system is a finite set of sites. There is an interaction between sites given by an Hamiltonian. A Gibbs measure is a probability on the finite set of configurations which is optimal with respect to some quantity. All the objects depend on a parameter β equal to the inverse of the temperature. Then, the issue is to define the possible accumulation points for the Gibbs measures as the number of sites goes to +∞. A phase transition occurs if, as moving the parameter β, this set of accumulation points admits a discontinuity for some β c .

In Ergodic Theory (on symbolic dynamics) one immediately considers the set of infinite configurations Σ. Instead of having an Hamiltonian one considers a potential ψ : Σ → R. We consider here the case where ψ is Hölder continuous. The Gibbs measure is an invariant measure, also called equilibrium state, which maximizes the pressure (see below). The pressure function is the pressure for β.ψ considered as a function of β. Then, a phase transition occurs if the pressure function is not analytic at some β c . It is noteworthy that in this setting, the number of equilibrium states is almost independent of the regularity of the pressure function (see [START_REF] Thaler | Estimates of the invariant densities of endomorphisms with indifferent fixed points[END_REF][START_REF] Leplaideur | Chaos: butterflies also generate phase transitions[END_REF]).

Roughly speaking, our dictionary identifies accumulation points of Gibbs measures in Probability with conformal measures in Ergodic Theory. Conformal measures are not Gibbs measures (within the Ergodic viewpoint) but are strongly related to them. More precisely, we show that for the Curie-Weiss model (see Theorem 1) and the Curie-Weiss-Potts model (see Theorem 3) the Gibbs measures converge to a convex combination of conformal measures. A (probabilistic) phase transition occurs when the number of conformal measures in the limit changes. This notion of phase transition is thus different from the one used in Ergodic Theory.

Although some of the convergences we prove here were already known (see [START_REF] Orey | Large deviations for the empirical field of Curie-Weiss models[END_REF]), we point out that the identification of the limit with ergodic quantities was not stated. We claim that the novelty here does not concern the convergence but the dictionary and identification/similitude/differences between Gibbs measures and phase transitions in Probability Theory or Ergodic Theory. We point out that this dictionary works well for the Curie-Weiss and Curie-Weiss-Potts models probably because for these models, the Hamiltonian turns out to be easily writable as a function of a Birkhoff sum. This is the core of Theorem 2.

At last, we mention works of Cioletti and Lopes (see [START_REF] Cioletti | Phase transitions in one-dimensional translation invariant systems: a Ruelle operator approach[END_REF][START_REF] Cioletti | Interactions, Specifications, DLR probabilities and the Ruelle Operator in the One-Dimensional Lattice[END_REF][START_REF] Cioletti | Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures[END_REF]). There, for different settings (non Hölder continuous potentials and not for the Curie-Weiss models), the connection between DLR-Gibbs measures (within the Statistical Mechanics viewpoint) and the conformal measures (within the Ergodic viewpoint) is also done. 1.2. Precise settings and results.

1.2.1. Ergodic and Dynamical settings. We consider a finite set Λ with cardinality bigger or equal to 2. It is called the alphabet. Then we consider the one-sided full shift Σ = Λ N over Λ. A point x in Σ is a sequence x 0 , x 1 , . . . (also called an infinite word) where the x i are in Λ. Most of the times we shall use the notation x = x 0 x 1 x 2 . . .. A x i ∈ Λ can either be called a letter, or a digit or a symbol.

The shift map σ is defined by

σ(x 0 x 1 x 2 . . .) = x 1 x 2 . . . .
The distance between two points x = x 0 x 1 . . . and y = y 0 y 1 . . . is given by

d(x, y) = 1 2 min{n, xn =yn} •
A finite string of symbols x 0 . . . x n-1 is also called a word, of length n. For a word w, its length is |w|. A cylinder (of length n) is denoted by [x 0 . . . x n-1 ]. It is the set of points y such that y i = x i for i = 0, . . . n -1. We shall also talk about n-cylinder instead of cylinder of length n.

If w is the word of finite length w 0 . . . w n-1 and x is a word, the concatenation wx is the new word w 0 w 1 . . . w n-1 x 0 x 1 . . .. For ψ : Σ → R continuous and β > 0, the pressure function is defined by ( 1)

P(βψ) := sup µ h µ + β Σ ψ dµ ,
where the supremum is taken among the set M σ (Σ) of σ-invariant probabilities on Σ and h µ is the Kolmogorov-Sinaï entropy of µ. The real parameter β is assumed to be positive because it represents the inverse of the temperature in statistical mechanics.

It is known that the supremum is actually a maximum and any measure for which the maximum is attained in (1) is called an equilibrium state for βψ. We refer the reader to [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF][START_REF] Ruelle | Thermodynamic formalism[END_REF] for basic notions on thermodynamic formalism in ergodic theory.

If ψ is Lipschitz continuous then the Ruelle-Griffith theorem (see [START_REF] Griffiths | Strict convexity (continuity) of the pressure in lattice systems[END_REF]) states that for every β, there is a unique equilibrium state for βψ, which is denoted by µ βψ . It is ergodic and it shall be called the dynamical Gibbs measure (DGM for short 1 ). It is the unique σ-invariant probability measure which satisfies the property that for every x = x 0 x 1 . . . and for every n,

(2) e -C β ≤ µ βψ ([x 0 . . . x n-1 ]) e β.Sn(ψ)(x)-nP(βψ) ≤ e C β
, where C β is a positive real number depending only on β and ψ (but not on x or n), and S n (ψ) stands for ψ

+ ψ • σ + . . . + ψ • σ n-1 .
With these settings, the βψ-conformal measure is the unique probability measure such that for every x and for every n,

(3) ν βψ ([x 0 . . . x n-1 ]
) = e βSn(ψ)(x 0 ...x n-1 y)-nP(βψ) dν βψ (y).

A precise (and more technical) definition of conformal measure is given in page 19.

Again, we refer the reader to [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF][START_REF] Ruelle | Thermodynamic formalism[END_REF] to see the connection between conformal measures and DGM. We emphasize that in our settings, conformal measures and DGM are equivalent measures and one can obtain one from the other.

If the choice of ψ is clear we shall drop the ψ and write µ β , ν β and P(β).

1.2.2.

The Curie-Weiss model. Probabilistic settings 1 and results. We consider the case Λ = {-1, +1}; Σ will be denoted by Σ 2 .

If ω 0 . . . ω n-1 is a finite word, we set (4)

H n (ω) := - 1 2n n-1 i,j=0 ω j ω i .
It is called the Curie-Weiss Hamiltonian. The empirical magnetization for ω is

m n (ω) := 1 n n-1 j=0 ω j . Then we have (5) H n (ω) = - n 2 (m n (ω)) 2 .
We denote by P := ρ ⊗N the product measure on Σ 2 , where ρ is the uniform measure on {-1, 1}, i.e. ρ({1}) = ρ({-1}) = 1 2 , and we define the probabilistic Gibbs measure (PGM for short) µ n,β on Σ 2 by [START_REF] Cioletti | Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures[END_REF] µ n,β (dω) := e -βHn(ω) Z n,β P(dω),

where Z n,β is the normalization factor

Z n,β = 1 2 n ω , |ω |=n e -βHn(ω ) .
Note that µ n,β can also be viewed as a probability defined on Λ n .

The measure P is a Bernoulli measure and is σ-invariant. In Ergodic Theory it is usually called the Parry-measure (see [START_REF] Parry | Zeta functions and the periodic orbit structure of hyperbolic dynamics[END_REF]) and turns out to be the unique measure with maximal entropy. With our previous notations it corresponds to the DGM µ 0 .

If

µ n,β w -→ n→+∞    µ 0 if β ≤ 1, 1 2 µ + 2βξ β + µ - 2βξ β if β > 1.
Remark 1. Actually µ n,β converges towards 1 2 µ + 2βξ β + µ - 2βξ β for every β > 0 since we shall see that for β ≤ 1 we have ξ β = 0, and it is clear that µ + 0 = µ - 0 = µ 0 = ρ ⊗N .

We refer to [START_REF] Ellis | Entropy, large deviations, and statistical mechanics[END_REF], sections IV.4 and V.9, for a discussion of the Curie-Weiss model and historical references (see also [START_REF] Rassoul | A course on large deviations with an introduction to Gibbs measures[END_REF], section 3.4). In Theorem IV.4.1 it is proved, using large deviations, that the sequence of image probabilities Q n defined by [START_REF] Ellis | Entropy, large deviations, and statistical mechanics[END_REF] Q n (A) := µ n,β (m n ∈ A) converges to an atomic measure as n goes to +∞. By using this theorem, Orey ([19], Corollary 1.2) proved by a nice simple probabilistic argument the weak convergence of µ n,β towards an explicit atomic measure. We were not aware of the work of Orey when we proved our theorem 1. We think it is still interesting because it makes the link with dynamical Gibbs measures, and furthermore our proof is direct and does not use large deviations.

We emphasize the equality [START_REF] Ellis | Limit theorems for sums of dependent random variables occurring in statistical mechanics[END_REF] m n (ω

) := 1 n S n (1 1 [+1] -1 1 [-1] )(ω)
which shows that m n can be written as a Birkhoff mean of a continuous function.

A consequence of ( 9) is that (5) can be rewritten under the form

H n (ω) = - n 2 1 n S n (ψ)(ω) 2 ,
where

ψ := 1 1 [+1] -1 1 [-1]
. We are thus led to investigate about results similar to those of Theorem 1 but for a more general Hamiltonian. From that point we discovered a very strange result: to export the Curie-Weiss Model in Ergodic Theory, one should focus on the invariant measures µ which maximize

h µ + β 2 ψ dµ 2
instead of measures which maximize h µ + β ψ dµ.

Theorem 2. Convergence for a more general Hamiltonian

Let ψ be a locally constant function 2 on Σ 2 . Let H n be the Hamiltonian defined by

H n (ω) = - n 2 1 n S n (ψ)(ω) 2 .
Let µ n,β be the PGM defined by [START_REF] Cioletti | Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures[END_REF] with this new value for H n . Then, for each β > 0,

(1) there are finitely many real numbers t 1 , . . . t J , and J depends on β such that the DGM µ βt j 's associated to βt j ψ are the unique measures which maximize

h µ + β 2 ψ dµ 2 .
(2) As n goes to +∞, µ n,β converges weakly to a convex combination of the conformal measures ν βt j 's associated to βt j ψ.

From this theorem it makes sense to define a new concept of pressure as

P 2 (βψ) := max h µ + β 2 ψ dµ 2 ,
and to study measures which realize this maximum. This quantity shall be referred to as the quadratic pressure. Note that upper semi-continuity for entropy shows that the maximum is well defined. The function β → P 2 (βψ) is obviously convex (thus continuous). Theorem 1 shows that it can be piecewise analytic and that the number of measures which realize the maximum may change with respect to β.

Remark 2. We emphasize that Theorem 1 is a particular case of Theorem 2 with

ψ = 1 1 [+] -1 1 [-]
. Note that for this particular case, the DGM is also the conformal measure.

2 That is ψ is of the form ψ = k j=0 a j 1 1 Cj where the C j 's are disjoint cylinders 1.2.3. Some consequences of Theorem 2. Several questions arise from Theorem 2. Actually, the main goal for Thermodynamic Formalism is to furnish a way to select one (or finitely many) invariant ergodic measure via the variational principle (see [START_REF] Baraviera | Ergodic optimization, zero temperature limits and the max-plus algebra[END_REF]). In terms of mathematics, it does not make neither more nor less sense to select measures which maximize

h µ + β ψ dµ instead of h µ + β 2 ψ dµ 2 .
Therefore, we are naturally led to redo the Thermodynamic Formalism for all known Dynamical Systems but inquiring for measures µ which maximize

h µ + β 2 ψ dµ 2 .
In particular, a natural question is to inquire whether Theorem 2 holds for any sufficiently regular potential ψ and not only the locally constant ones. We point out that our proof cannot be easily adapted to that more general case. Moreover, the usual tool in Thermodynamic Formalism -that is the Transfer operator -does not seem to be well adapted to this question.

For more geometric dynamical systems, one usually considers or studies the special class of physical or/and SRB-measures. These measures are usually considered as the most natural ones with the measures of maximal entropy. It is clear that measures of maximal entropy also maximize h µ + ψ dµ 2 for ψ ≡ 0. A natural question is thus to know if SRB-measures can also be characterized as measures maximizing

h µ + ψ dµ 2 .
1.2.4. The Curie-Weiss-Potts model. Probabilistic settings 2 and result. The Curie-Weiss-Potts model will be for Λ = {θ 1 , . . . , θ q } with q > 2. In that case we shall write Σ q instead of Σ.

The Curie-Weiss-Potts Hamiltonian is defined for a finite word

ω = ω 0 • • • ω n-1 by (10) H n (ω) := - 1 2n n-1 i,j=0 1 1 ω j =ω i .
We define the vector

L n (ω) = (L n,1 (ω), • • • , L n,q (ω))
where

L n,k (ω) = n-1 i=0 1 1 ω i =θ k
is the number of digits of ω which take the value θ k , so that we can write

n-1 i,j=0 1 1 ω j =ω i = q k=1 n-1 i=0 1 ω i =θ k 2 = L n (ω) 2 ,
where • stands for the euclidean norm on R q .

We denote by P := ρ ⊗N the product measure on Σ q , where ρ is the uniform measure on Λ, i.e. ρ = 1 q q k=1 δ θ k , and we define the probabilistic Gibbs measure µ n,β on Σ q by [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF] µ n,β (dω

) := e -βHn(ω) Z n,β P(dω) = e β 2n Ln(ω) 2 Z n,β P(dω),
where Z n,β is the normalization factor

Z n,β = 1 q n ω , |ω |=n e β 2n Ln(ω ) 2 .
Now we can state the analog of Theorem 1.

Theorem 3. Weak convergence for the CWP model

For 1 ≤ k ≤ q, b ∈ R, let µ k b be the dynamical Gibbs measure for b1 1 [θ k ] . Let β c = 2(q-1) log(q-1) q-2
. For 0 < β < β c set s β = 0 and for β ≥ β c let s β be the largest solution of the equation [START_REF] Feng | Lyapunov spectrum of asymptotically sub-additive potentials[END_REF] s = e βs -1 e βs + q -1 .

Then,

(13) µ n,β w -→ n→+∞                ρ ⊗N if 0 < β < β c , 1 q q k=1 µ k βs β if β > β c , A µ 1 0 + B q k=1 µ k βcs βc A + qB if β = β c , with A = 1 -βc q(q-1) q-2 2 and B = 1 -βc q q-2 2 .
Remark 3. Actually µ n,β converges towards 1 q q k=1 µ k βs β for every β = β c since s β = 0 for β < β c , and it is clear that

µ k 0 = ρ ⊗N for each 1 ≤ k ≤ q.
We refer to [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF] for a discussion of the Curie-Weiss-Potts model and historical references. Orey ([19], Theorem 4.4) mentions the weak convergence of µ n,β towards an explicit atomic measure, but he makes a mistake concerning the case β = β c , as pointed out in [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF].

It is highly probable that a similar result to Theorem 2 holds for the Curie-Weiss-Potts model. Note that as for Theorem 1, in the settings of Theorem 3 conformal and Gibbs measures are equal.

1.3. Plan of the paper. The paper is composed as follows.

In Section 2 we prove Theorem 1. In Section 3 we prove Theorem 3. Both proofs are very similar and are based on the convergence of µ n,β (C) where C is a cylinder in Σ.

Theorem 2 is proved in Section 4. The proof is similar to the ones of Theorem 1 and 3. The main difference is that the auxiliary function is more delicate to control.

For that, we need to use the Transfer Operator.

Proof of Theorem 1

To prove the convergence of µ n,β towards µ, it is enough to show that for every cylinder C,

(14) lim n→∞ µ n,β (C) = µ(C).
First we justify that ϕ I admits a unique maximum in [0, 1] and use this point to get convergence for µ n,β (C), where C is any cylinder. In the second subsection we show that this limit is equal to the right convex combinations of DGM's.

2.1. The auxiliary function ϕ I and limit for µ n,β . We recall that we set ϕ I (x) := log(cosh(βx)) -β 2 x 2 . Lemma 2.1. Maxima for ϕ I The function ϕ I attains its maximum on R + at a unique point ξ β which is the unique non-negative solution of the equation tanh

(βx) = x. If β ≤ 1, then ξ β = 0. Proof. Note that ϕ I (x) = β (tanh(βx) -x) and ϕ I (x) = β(β -1 -β tanh 2 (βx)). If β ≤ 1, ϕ I is non-positive,
thus ϕ I decreases and ϕ I (0) = 0 yields that ϕ I is a decreasing function. The maximum is then attained for ξ β = 0.

If β > 1, then ϕ I is positive and then negative, which yields that ϕ I is first an increasing and then a decreasing function. Note that ϕ I (0) = 0 and ϕ I (1) < 0, which shows that ϕ I is positive on some interval ]0, ξ β [ with 0 < ξ β < 1 and negative on ]ξ β , +∞[. Consequently, ϕ I reaches its maximal value on R at the points ±ξ β defined by [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] tanh(βξ β ) = ξ β .

Now we are ready to compute the limit of a fixed cylinder. Let ω = ω 0 . . . ω p-1 be a finite word of length p. We denote by S p (ω) = p-1 i=0 ω i the sum of the p digits of ω.

Lemma 2.2. Limit of the measure of a fixed cylinder

(16) lim n→∞ µ n,β ([ω 0 . . . ω p-1 ]) =      1 2 p if β ≤ 1, 1 2 (f (ξ β ) + f (-ξ β )) if β > 1,
where f (y) = e βySp(ω) (e βy + e -βy ) p .

Proof. For big n,

[ω] = [ω 0 . . . ω p-1 α p . . . α n-1 ] = α, |α|=n-p [ωα],
and

µ n,β ([ω]) = α, |α|=n-p µ n,β ([ωα]) = α, |α|=n-p e -βHn(ωα)
α, |α|=n e -βHn(α) .

For any word x = x 0 x 1 x 2 . . . and any n ∈ N * , we denote by S n (x) = n-1 i=0 x i the sum of the n first digits of x. Then We use the equality

e a 2 = 1 √ 2π +∞ -∞ e -x 2 2 + √ 2ax dx,
sometimes called the Hubbard-Stratonovich transformation ( [START_REF] Hubbard | Calculation of partition functions[END_REF], [START_REF] Stratonovič | A method for the computation of quantum distribution functions[END_REF]), to compute the following. X i ) where the X i 's are independent identically distributed with law P(X i = -1) = P(X i = 1) = 1 2 , and we deduce that α, |α|=n-p e βyS n-p (α) = (e βy + e -βy ) n-p = 2 n cosh(βy) n (e βy + e -βy ) p .

α, |α|=n-p e -βHn(ωα) = 1 √ 2π +∞ -∞ e -x 2 2 α e √ β n xSn(ωα) dx = 1 √ 2π +∞ -∞ e -x 2 2 + √ β n xSp(ω) α e √ β n xS n-p (α) dx = √ nβ √ 2π +∞ -∞ e -n
Eventually we obtain [START_REF] Keller | Equilibrium states in ergodic theory[END_REF] α, |α|=n-p

e -βHn(ωα) = 2 n √ nβ √ 2π +∞ -∞
e nϕ I (y) f (y) dy.

Similarly, p = 0 yields

α, |α|=n e -βHn(α) = 2 n √ nβ √ 2π +∞ -∞
e nϕ I (y) dy, therefore we obtain that

(18) µ n,β ([ω]) = +∞ -∞ e nϕ I (y) f (y) dy +∞ -∞ e nϕ I (y) dy .
We recall that the Laplace method shows that if ϕ I vanishes on a single point ξ in the interior of the interval I and if ϕ I (ξ) < 0 and f (ξ) = 0, then

I e nϕ I (y) f (y)dy ∼ n→∞ √ 2π |ϕ I (ξ)| e nϕ I (ξ) f (ξ)n -1/2 . (19)
We remind that

u n ∼ n→+∞ v n means u n = v n (1 + ε(n)) with lim n→+∞ ε(n) = 0.
If β < 1: we may consider I = R and ξ = ξ β = 0. We thus get ϕ I (0)) = β(β -1), f (0) = 1 2 p , and

µ n,β ([ω]) ∼ n→∞ f (0) 1 = 1 2 p .
If β = 1: in this case ϕ I (0)) = 0 but the Laplace method still works if we consider the least integer k such that ϕ (k) I (0) = 0. We do not need to calculate it because we have as in the preceding case that

e nϕ I (y) f (y)dy ∼ n→∞ √ 2π |ϕ I (ξ β )| e nϕ I (ξ β ) (f (ξ β ) + f (-ξ β )) n -1/2 , which yields µ n,β ([ω]) ∼ n→∞ f (ξ β ) + f (-ξ β ) 2 .
2.2. Identification of the limit as a convex combination of DGM's. First, we point out that Lemma 2.2 yields lim n→+∞ µ n,β = µ 0 if β ≤ 1. We thus have to deal with the case β > 1.

Let us first compute the dynamical Gibbs measures µ + b and µ - b . We denote by L p,+ (ω) := p-1 k=0 1 ω k =1 the number of digits of ω which take the value 1, and similarly L p,-(ω) := p-1 k=0 1 ω k =-1 is the number of digits which take the value -1. Proof.

-p log p -(1 -p) log(1 -p) + bp = sup q∈[0,1] (-q log q -(1 -q) log(1 -q) + bq) .
It is easy to show that p = e b 1+e b , and then

µ + b ([ω]) = e bL p,+ (ω) (e b + 1) p .
Exchanging the roles of +1 and -1 gives the equality

µ - b ([ω]) = e bL p,-(ω) (e b + 1) p .
We recall that S p (ω

) = p-1 k=0 ω k = L p,+ (ω) -L p,-(ω) = 2L p,+ (ω) -p = p -2L p,-(ω). Then f (ξ β ) = e βξ β Sp(ω) (e βξ β + e -βξ β ) p = e βξ β (2L p,+ (ω)-p) (e βξ β + e -βξ β ) p = e 2βξ β L p,+ (ω) (e 2βξ β + 1) p .
Similarly we have f (-ξ β ) = e 2βξ β L p,-(ω) (e 2βξ β + 1) p and replacing these values in [START_REF] Hubbard | Calculation of partition functions[END_REF] we get

lim n→+∞ µ n,β ([ω]) = 1 2 ( µ + 2βξ β ([ω]) + µ - 2βξ β ([ω])),
and the theorem is proved.

Proof of Theorem 3

The proof follows the same lines as the proof of Theorem 1. In the first step we use an auxiliary function ϕ P . Note that the function was already studied by Ellis and Wang in [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF]. Then we deduce that µ n,β (C) converge for any cylinder C. In the second step we identify the limit as the relevant convex combination of dynamical measures.

3.1. Auxiliary function ϕ P and convergence for µ n,β . We shall need the function ϕ P defined on R q by (21)

ϕ P (z) = - β 2 z 2 + log q k=1 e βz k .
This function attains its maximum on R q since ϕ P (z) ≤ -c z 2 as z tends to ∞. We recall Theorem 2.1 of [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF], which describes precisely the global maximum points of ϕ P .

Theorem 4. (Ellis Wang [11])

Let β c = 2(q-1) log(q-1) q-2 . For 0 < β < β c set s β = 0 and for β ≥ β c let s β be the largest solution of the equation [START_REF] Ruelle | Statistical mechanics[END_REF] s = e βs -1 e βs + q -1 .

The function β → s β is strictly increasing on the interval [β c , +∞[, s(β c ) = q-2 q-1 , and lim β→∞ s β = 1.

Denote by φ the function from

[0, 1] into R q defined by φ(s) = 1 + (q -1)s q , 1 -s q , • • • , 1 -s q ,
the last (q -1) components all equal 1-s q . Let K β denote the set of global maximum points of the symmetric function ϕ P . Define

ν 0 = φ(0) = 1 q , • • • , 1 q . For β ≥ β c , define ν 1 (β) = φ(s β ) and let ν i (β), i = 2, • • • , q denote the
points in R q obtained by interchanging the first and ith coordinates of ν 1 (β). Then

K β =      {ν 0 } for 0 < β < β c , {ν 1 (β), ν 2 (β), • • • , ν q (β)} for β > β c , {ν 0 , ν 1 (β c ), ν 2 (β c ), • • • , ν q (β c )} for β = β c .
For β ≥ β c the points in K β are all distinct.

We fix a finite word ω = ω 0 • • • ω p-1 of length p and we compute the limit of µ n,β ([ω]).

Lemma 3.1.

lim n→∞ µ n,β ([ω]) =                    1 q p if β < β c , 1 q 1 (e βs β + q -1) p q k=1 e βs β L p,k (ω) if β > β c , A q p + B (e βs β +q-1) p q k=1 e βcs βc L p,k (ω) A + qB if β = β c .
Proof. The proof is very much the same as the proof of Lemma 2.2, but the Hamiltonian has changed and we are in dimension q instead of dimension one. We want to evaluate the limit of

µ n,β ([ω]) = α, |α|=n-p µ n,β ([ωα]) = α, |α|=n-p e β 2n Ln(ωα) 2 α, |α|=n e β 2n Ln(α) 2 .
With the help of the identity ( 23)

e u 2 = 1 (2π) q/2 R q exp - 1 2 y 2 + √ 2
) 2 = 1 (2π) q/2 R q exp - 1 2 y 2 + β n y, L p (ω) + (n -p) log q k=1 e √ β n y k dy.
Now we make the change of variable βz = β n y, and we obtain [START_REF] Sinai | Gibbs measures in ergodic theory[END_REF] α, |α|=n-p

e β 2n Ln(ωα) 2 = nβ 2π q/2
R q e nϕ P (z) f (z) dz, where ϕ P was defined in ( 21) and f is defined on R q by ( 25)

f (z) = exp β z, L p (ω) -p log q k=1 e βz k .
Similarly, p = 0 yields

α, |α|=n e β 2n Ln(α) 2 = nβ 2π q/2
R q e nϕ P (z) dz, hence = R q e nϕ P (z) f (z) dz R q e nϕ P (z) dz .

µ n,β ([ω]) = α,
We denote by Dϕ P (z), respectively H(z), the gradient, respectively the Hessian matrix, of ϕ P at z. It is proved in Proposition 2.2 of [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF] that the Hessian matrix of ϕ P is negative definite at each global maximum point of ϕ P . Now if Dϕ P vanishes at a single point z 0 in an open set O of R q , if H(z 0 ) is negative definite and if f (z 0 ) = 0, then we know by Laplace's method that

0 e nϕ P (z) f (z) dz ∼ n→∞ (2π) q/2 f (z 0 )e nϕ P (z 0 ) n q/2 |det H(z 0 )| .
If 0 < β < β c : according to Theorem 4, ϕ P attains its maximum at the unique point ν 0 so applying Laplace's method yields

µ n,β ([ω]) ∼ n→∞ f (ν 0 ) 1 = 1 q p .
If β > β c : Theorem 4 states that ϕ P attains its maximum at exactly q points ν i (β), i = 1, • • • , q, where ν i (β), i = 2, • • • , q is obtained by interchanging the first and ith coordinates of ν 1 (β). Due to the symmetry of the function

ϕ P it is clear that det H(ν i ) = det H(ν 1 ), i = 2, • • • , q. Considering a family of disjoint open sets (O i ) 1≤i≤q such that O i contains ν i and R q = ∪ q i=1 O i ∪N ,
where N is a set of measure zero, Laplace's method yields

µ n,β ([ω]) ∼ n→∞ 1 q q i=1 f (ν i ).
Recall that

f (ν i ) = e β ν i ,Lp(ω) q k=1 e βν i k p with ν i k =        1 -s β q if k = i, 1 + (q -1)s β q if k = i.
As q k=1 L p,k (ω) = p it is easily seen that ( 26)

e β ν i ,Lp(ω) = exp βp(1 -s β ) q + βs β L p,i (ω) .
As ν i is a critical point of ϕ P and ∂ϕ P ∂z i (z) = βe βz i q k=1 e βz k -βz i , we know that [START_REF] Thaler | Estimates of the invariant densities of endomorphisms with indifferent fixed points[END_REF] q k=1 e βν i k = e βν i j

ν i j = q 1 -s β e β(1-s β ) q
.

Putting together ( 26) and ( 27) we obtain

f (ν i ) = 1 -s β q p e βs β L p,i (ω) ,
which can also be written

(28) f (ν i ) = 1 (e βs β + q -1) p e βs β L p,i (ω)
since s β is solution of the equation [START_REF] Feng | Lyapunov spectrum of asymptotically sub-additive potentials[END_REF]. Therefore

µ n,β ([ω]) ∼ n→∞ 1 q
1 (e βs β + q -1) p q i=1 e βs β L p,i (ω) .

If β = β c : the function ϕ P admits exactly q + 1 maximun points ν i (β), i = 0, • • • , q but det H(ν 0 ) = det H(ν 1 ), therefore Laplace's method yields

(29) µ n,β ([ω]) ∼ n→∞ |det H(ν 0 )| -1/2 f (ν 0 ) + |det H(ν 1 )| -1/2 q i=1 f (ν i ) |det H(ν 0 )| -1/2 + q |det H(ν 1 )| -1/2 .
In the proof of Proposition 2.2 of [START_REF] Ellis | Limit theorems for the empirical vector of the Curie-Weiss-Potts model[END_REF] it is proved that H(ν 0 ) has a simple eigenvalue at β and an eigenvalue of multiplicity (q-1) at βq -1 (q-β) whereas H(ν 1 ) has simple eigenvalues at β and β -β 2 qab and an eigenvalue of multiplicity (q -2) at β -β 2 b, where a = q -1 (1

+ (q -1)s β ) and b = q -1 (1 -s β ). Recalling that s(β c ) = q-2 q-1 we deduce that |det H(ν 0 )| = β q c (1 -q -1 β c ) q-1 , |det H(ν 1 )| = β q c (1 -q -1 β c ) 1 - β c q(q -1) q-2 .
Reporting in (29) and recalling (28) we get the result.

3.2. Identification of the limit. We can already deduce from Lemma 3.1 that

µ n,β w -→ n→+∞ µ 0 if β < β c . Lemma 3.2. Computation for µ k b For k = 1, . . . , q, (30) µ k b ([ω]) = e bL p,k (ω) (e b + q -1) p .
Proof. The function b1 1 [θ k ] depends only on the zero coordinate, therefore the supremum in ( 1) is attained for the product measure (m k ) ⊗N , where the probability vector (m k j ) 1≤j≤q on Λ maximizes the quantity q j=1 p j log p j + bp k over all the probability vectors (p j ) 1≤j≤q on Λ, and is given by [START_REF] Keller | Equilibrium states in ergodic theory[END_REF]). The result is then clear.

m k k = e b e b +q-1 , m k j = 1 e b +q-1 if j = k (see for instance Example 4.2.2 of
The limit in ( 13) is now a direct consequence of the lemmas 3.1 and 3.2.

Proof of Theorem 2

We consider a function ψ of the form

ψ := i a i 1 1 C i ,
where the sum is finite, the C i 's are disjoint cylinders, and the a i 's are real numbers which are not all equal to zero. We can always assume that all the cylinders have the same length, say q.

In the following we assume that we have chosen an order on q-cylinders and we consider the vector a := (a 1 , . . . , a 2 q ). 4.1. Convergence of µ n,β . Let ω = ω 0 . . . ω p-1 be a finite word of length p. To fix the ideas, we assume that p is bigger than q. Note that if the convergence holds for any cylinder with length larger than q then it holds for any cylinder.

For n very big and x ∈ Σ 2 , S n (ψ)(x) involves digits x 0 , . . . , x n+q-2 . Then, we get

µ n,β ([ω]) = α, |α|=n-p+q-1 µ n,β ([ω 0 . . . ω p-1 α 0 . . . α n-p+q-2 ]).
For simplicity we write S n (ωα) for S n (ψ)(ωα) (which is well defined if |α| ≥ n -p + q -1). Again, using the Hubbard-Stratonovich transformation we get We point out the equality (32) S n (ωα) = S p-q+1 (ω) + S q-1 (σ p+1-q (ωα)) + S n-p (α).

This equality means that the first p + 1 -q terms in the Birkhoff sum S n (ωα) only involve the ω i 's, then come q -1 terms which involve the tail of ω and the head of α, and then comes the tail of the sum which only depends on α.

Let us define the operator L ξ , depending on a real or complex parameter ξ, by L ξ (T )(x) := e ξψ(-1x) T (-1x) + e ξψ(1x) T (1x).

Here x is a point in Σ 2 , T is a continuous function on Σ 2 , and we remind that ix, with i = ±1, is the infinite word ix 0 x 1 x 2 . . .. L ξ is the transfer operator L n-p β.z (e βzS q-1 (σ p+1-q (ω.)) )(α x).

We point out that the x on the right hand side of this equality is just there to make sense but the expression actually does not depend on x because |α | = q -1.

It is known (see [START_REF] Bowen | Equilibrium states and the ergodic theory of Anosov diffeomorphisms[END_REF]) that L ξ acts on continuous and Hölder continuous functions. Moreover on the space of Hölder continuous functions it is quasi-compact: its spectral radius, say λ(ξ), is a simple dominating eigenvalue associated to a positive eigenfunction, say H ξ . The adjoint operator acts on measures and it admits a unique eigenmeasure associated to the spectral radius, say ν ξ . Moreover H ξ dν ξ = 1 and for every continuous function T ,

lim n→+∞ max x∈Σ 2 λ(ξ) -n L n ξ (T )(x) - T dν ξ H ξ (x) = 0.
3 also called the Ruelle Perron Frobenius operator.

Inequalities between eigenvalues of matrices and coefficients yield λ(βt) ≤ 2e Aβt .

This shows that for t ≥ 0, ϕ OS (t) ≤ -β 2 t 2 + Aβt + log 2, and this upper bound goes to -∞ if t goes to +∞. On the other hand, ϕ OS (0) = log 2. Therefore, ϕ OS reaches its maximum on R + only on some compact set [0, κ(β)].

Furthermore, any t which realizes the maximum must be a critical point of the derivative. Note that ξ → λ(ξ) admits an analytic continuation in a complex neighborhood of R which prohibits to get infinitely many zeros of ϕ OS on the segment [0, κ(β)]. Proposition 4.2. Let β be fixed. Let (t i ) 1≤i≤J be the finitely many real numbers where ϕ OS reaches its maximum4 . Then, there exist finitely many positive b i , 1 ≤ i ≤ J such that J i=1 b i = 1 and such that µ n,β ([ω]) converges to

J i=1 b i ν βt i ([ω]).
Proof. From the definition of µ n,β we know that .

Using (31) and (33) we get

(35) µ n,β ([ω]) = |α|=q-1 +∞ -∞ e -n β 2 z 2 e βzS p-q+1 (ω) L n-p βz (T βz )(α) dz |α|=q-1 +∞ -∞ e -n β 2 z 2 L n βz (1 1)(α) dz .
We can rewrite the numerator as a sum of q terms of the form

+∞

-∞ e nϕ OS (βz) e βzS p-q+1 (ω)-p log λ(βz) L n-p βz λ(βz) n-p (T βz )(α) dz, and the denominator as a sum of q terms of the form

+∞ -∞ e nϕ OS (βz) L n βz λ(βz) n (1 1)(α) dz.
We want to use the Laplace method for these terms. Lemma 4.1 shows that the exponential term (in n) has only finitely many maximum. But we are not yet done and the problem comes from the last term in the integral which depends on n. Inequality (34) shows that this term converge as n goes to +∞ but the speed of convergence depends on z and z may goes to ±∞.

To deal with this fact, we recall that A is max |a i | and we notice that for every continuous T ,

||L βz (T )|| ∞ ≤ 2e Aβ|z| ||T || ∞ , therefore ||L βz (T βz )|| ∞ ≤ 2e qβ|z|A , and
|e -n β 2 z 2 e βzS p-q+1 (ω) L n-p βz (T βz )(α)| ≤ e n(-β 2 z 2 +β|z|A(p+1)+log 2) .
As -β 2 z 2 + |z|(βA(p + 1) + 1) + log 2 goes to -∞ as |z| goes to +∞, we can deduce that there exists κ(β) > 0 such that for every n ≥ 2, (36)

|z|≥κ(β) e -n β 2 z 2 e βzS p-q+1 (ω) L n-p βz (T βz )(α) dz ≤ e -nκ(β) .
We choose κ(β) big enough so that all the points t j , 1 ≤ j ≤ J, where ϕ OS reaches its maximum, are in the interior of the interval [-κ(β), κ(β)]. Then we can write the integral over the segment [-κ(β), κ(β)] as a finite sum of integrals over segments [a j , b j ] where each segment [a j , b j ] contains exactly one of the points t j , 1 ≤ j ≤ J.

We state the following lemma, which is an immediate adaptation of the Laplace method.

nϕ(x) f n (x) dx ∼ π 2|ϕ (c)| e nϕ(c) f (c)n -1/2 .
We apply this lemma on every [a j , b j ] to the functions f n defined by

f n (z) = e βzS p-q+1 (ω)-p log λ(βz) L n-p βz λ(βz) n-p (T βz )(α).
Thanks to (34) the functions f n converge uniformly on [a j , b j ] to f defined by

f (z) = e βzS p+1-q (ω) λ(βz) p e βzS q-1 (σ p+1-q (ωx)) dν βz (x) .H βz (α) = 1 λ(βz) p e βzSp(ωx) dν βz (x) .H βz (α) = 1 λ(βz) p L p βz (1 1 [ω] )(x) dν βz (x) .H βz (α) = H βz (α)ν βz ([ω]).
Putting together (36) and the result of Lemma 4.3 applied to every [a j , b j ] we obtain that the numerator of µ nβ ([ω]) in ( 35) is equivalent to

|α|=q-1 J j=1 π 2|ϕ OS (t j )| e nϕ OS (t j ) H βt j (α)ν βt j ([ω])n -1/2 .
Similarly, the denominator of µ nβ ([ω]) in ( 35) is equivalent to

|α|=q-1 J j=1 π 2|ϕ OS (t j )| e nϕ OS (t j ) H βt j (α)n -1/2 .
We then get that µ nβ ([ω]) is equivalent to

J j=1 |α|=q-1 H βt j (α) |ϕ OS (t j )| ν βt j ([ω]) J j=1 |α|=q-1 H βt j (α) |ϕ OS (t j )| . Let us set b i := α,|α|=q-1 H βt i (α) |ϕ OS (t i )| j,α,
= g d µ t 0 ,
where µ t 0 is the DGM for t 0 g.

We recall also that for g = ψ = 2 q i=1 a i 1 1 C i , P(βtψ) = log λ(βt), where λ(βt) is the spectral radius for the matrix M (βta).

4.2.2.

Maximizing the quadratic pressure. There are 2 q q-cylinders in Σ 2 . If µ and ν are invariant measures, then µ ≈ ν ⇐⇒ µ([ω 0 . . . ω q-1 ]) = ν([ω 0 . . . ω q-1 ]) for every q-cylinder.

defines an equivalence relation over the set of invariant probabilities. If µ ≈ ν then ψ dµ = ψ dν holds because ψ is constant on q-cylinders.

Note that h µ + β 2 ψ dµ 2 is maximal among elements of the same class for ≈ if and only if h µ is maximal.

Because we are considering probability measures, it is sufficient to know µ(C i ) for all i < 2 q to know all of the measure µ. If y i , i = 1, . . . 2 q -1 are non-negative real numbers such that y i ≤ 1, then we denote by ≈ y , the class of measures such that µ(C i ) = y i for all i < 2 q . Now, we state a lemma that is an adapted version of results from [START_REF] Barreira | Higher-dimensional multifractal analysis[END_REF] (th. 7, 8) and [START_REF] Feng | Lyapunov spectrum of asymptotically sub-additive potentials[END_REF] (th 4.8). These papers focus on multifractal analysis and they have different ways to state the result we need here. It is not clear to us if their statements can be directly applied in our case. Lemma 4.4. Let y be such that ≈ y is non-empty. Then max {h µ , µ ∈≈ y } ≤ inf z∈R 2 q -1 , z =0

P z i 1 1 C i - z i y i
Proof. Note the immediate equality: inf t∈R + , z∈S 2 q -2 P(t

z i 1 1 C i ) -t z i y i = inf z∈R 2 q -1 , z =0 P z i 1 1 C i - z i y i
For our proof, we prefer to use the left hand side characterization. Let us pick z ∈ S 2 q -2 . First we assume that z i y i = max z i 1 1 C i dµ, where the maximum is taken among all invariant measures. Then, it is well known (see [START_REF] Baraviera | Ergodic optimization, zero temperature limits and the max-plus algebra[END_REF]) that t → P(t z i 1 1 C i ) -t z i y i is a convex decreasing function which converges to the residual entropy:

lim t→+∞ P(t z i 1 1 C i ) -t z i y i = max h µ , z i 1 1 C i is maximal .
Clearly the residual entropy is bigger than max {h µ , µ ∈≈ y }.

Let us now assume that the class ≈ y is not included into the set of maximizing measures for z i 1 1 C i . Then lim t→+∞ P(t z i 1 1 C i ) -t z i y i = +∞ and the convex function admits a unique minimum on R + .

We immediately get the following sequence of inequalities:

h y + β 2 t 2 ≤ log λ(βt) -βta 2 q -βt (a i -a 2 q )y i + β 2 t 2
≤ log λ(βt) + β 2 t 2 -βt ψ|y

≤ log λ(βt) + β 2 t 2 -βt 2 = ϕ OS (t),
where the last inequality uses t = ψ|y .

Lemma 4.6. If t is a critical point for ϕ OS , we consider the DGM µ tβψ associated to tβψ. We also set y i = µ tβψ (C i ) for 1 ≤ i ≤ 2 q and y := (y 1 , . . . y 2 q -1 ).

Then, t = ψ d µ tβψ and h y + β 2 t 2 = ϕ OS (t).

Proof. The DGM µ tβψ has pressure log λ(βt). Moreover Equality (37) yields

λ (βt) λ(βt) = ψ d µ βtψ .
As t is a critical point for ϕ OS we have βt = β. λ (βt) λ(βt) , and then t = ψ d µ tβψ holds. We deduce that P(βtψ) -βt ψ|y = log λ(βt) -βt 2 = ϕ OS (t) -

β 2 t 2 .
Now by definition of the DGM we know that P(βtψ) -βt ψ|y = h µ tβψ which is smaller than h y by definition of this last term, therefore

ϕ OS (t) ≤ h y + β 2 t 2 .
The reverse inequality holds true thanks to Lemma 4.5, which finishes the proof.

We can now finish the proof of Theorem 2. We have seen that the PGM µ β,n converges to j b j ν βt j , where the t j 's are the maxima for ϕ OS .

For these t j 's, Lemmas 4.5 and 4.6 show that

h µ + β 2 ψ dµ 2 ≤ ϕ OS (t j ),
with equality if µ is one of the DGM's, µ βt k ψ .

e

  -βHn(ωα) = e √ β 2n Sn(ωα) 2 and S n (ωα) = S p (ω) + S n-p (α).

βy 2 2

 2 +βySp(ω) α e βyS n-p (α) dy where we made the change of variable βy := β n x. Now we notice that the sum α, |α|=n-p e βyS n-p (α) is equal to 2 n-p E(e βy n-p-1 i=0

Re

  nϕ I (y) f (y)dy ∼ n→∞ 1 2 p R e nϕ I (y) dy, therefore we still have µ n,β ([ω]) ∼ n→+∞ 1 2 p . If β > 1: we may consider two intervals R -and R + and ξ = ±ξ β . Then we get +∞ -∞

Lemma 2 . 3 .

 23 Computation for µ ± b (20) µ + b ([ω]) = e bL p,+ (ω) (e b + 1) p and µ - b ([ω]) = e bL p,-(ω) (e b + 1) p .

  y, u dy, and noticing that L n (ωα) = L p (ω) + L n-p (α), we write α, L n-p (α) dy. It is easily seen that α,|α|=n-p e √ β n y,L n-p (α) =

  |α|=n-p e β 2n Ln(ωα) 2 α, |α|=n e β 2n Ln(α) 2

e -n β 2 z 2 αe

 2 βzSn(ωα) dz, where we have made the change of variable βz = β n x.

3

 3 associated to the potential ξ.ψ. We notice that for any m ∈ N * , L m ξ (T )(x) := α, |α|=m e ξSm(αx) T (αx). Now, we write α = α α , with |α | = n -p and |α | = q -1. Using the equality (32) it is easy to check that (33) α e βzSn(ωα) = e βzS p-q+1 (ω) α |α |=q-1

Lemma 4 . 3 .

 43 Let ϕ : [a, b] → R a function of class C 2 , with ϕ vanishing on a single point c in ]a, b[ and ϕ (c) < 0. Let (f n ) n≥1 , f some continuous functions from [a, b] to R such that f n converges to f uniformly on [a, b], and f (c) = 0. Then as n → ∞ b a e

  P n , P are probability measures on the Borel sets of a metric space S, we say that P n converges weakly to P if S f dP n → S f dP for each f in the class C b (S) of bounded, continuous real functions f on S. In this case we write P n

	w -→ n→+∞	P .
	Our first result concerns the weak convergence of the measures µ n,β .
	Theorem 1. Weak convergence for the CW model
	Let ξ β be the unique point in [0, 1] which realizes the maximum for ϕ I (x) := log(cosh(βx))-
	β 2 x 2 . Let µ + b and µ -b be the dynamical Gibbs measures for b1 1 [+1] and b1 1 [-1] respec-
	tively. Then
	(7)

  End of the Proof of Theorem 2. 4.2.1. Some properties for Pressure function. We recall here some properties for the pressure function. If g is Hölder continuous on Σ 2 then t → P(tg) is convex thus continuous. The Ruelle theorem also yields

	(37)	∂ ∂t	P(tg) t=t 0

|α|=q-1 H βt j (α) |ϕ OS (t j )| . Then the b i 's are positive, they satisfy i b i = 1, and µ n,β ([ω]) converges to i b i ν βt i ([ω]).

4.2.

We prefer the adjective "dynamical" instead of "ergodic" to avoid the discussion if an ergodic Gibbs measure is ergodic or not.

We emphasize that the t i 's depend on β but we drop the index β to get lighter notations.

Remark 4. The measure ν ξ is exactly the conformal measure associated to ξψ.

We refer the reader to [START_REF] Hennion | Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness[END_REF] for properties of quasi-compact operators ,in particular regularity of the maximal eigenvalue and the eigenvectors with respect to ξ.

Furthermore, L ξ acts on functions constant on (q-1)-cylinders because ψ is constant on q-cylinders. In that case we can give a simple characterization for λ(ξ). First, we order the (q -1)-cylinders, say K 1 , . . . , K 2 q-1 , and the q-cylinders, say C 1 , . . . , C 2 q . Let z = (z 1 , . . . , z 2 q ) be a vector in R 2 q . We consider the 2 q-1 × 2 q-1 matrix M (z) whose entries are defined as follows: From the (q -1)-cylinder

] and [ω i 1 , . . . , ω i q-2 , -1], corresponding respectively to say K j 1 (i) and K j 2 (i) . We can also define two q-cylinders C k 1 (i) = [ω i 0 , . . . , ω i q-2 , +1] and

A simple computation shows that

Let us call E q-1 the vector space of functions constant on (q -1)-cylinders. The set

} is a basis of E q-1 , which can be identified with the canonical basis (e i ) 1≤i≤2 q-1 of R 2 q-1 . With this identification we get that the restriction of L ξ to E q-1 is a linear operator whose matrix is exactly the transpose t M (ξ.a) of M (ξ.a). Therefore, λ(ξ) is the dominating eigenvalue for M (ξ.a). We emphasize that this matrix is aperiodic and then the Perron Frobenius theorem on matrices holds. Furthermore, H ξ is a left eigenvector for M (ξ.a) associated to λ(ξ).

Let us denote by T ξ the function x → e ξS q-1 (σ p+1-q (ωx)) . This function T ξ belongs to E q-1 , and

where ε(ξ) is continuous and positive (in ξ) and C(ξ) does only depend on ξ. Lemma 4.1. Let ϕ OS be defined be ϕ OS (t) = -β 2 t 2 + log λ(βt). Then, for each β, ϕ OS attains its maximum on R at finitely many points.

Proof. Recall that A = max |a i |. We do the case t ≥ 0, the case t ≤ 0 being the same with -A instead of A.

If this minimum is reached at t = 0, then its value is h = log 2 which is bigger than any metric entropy.

The last case holds if the minimum is reached at a critical point, say t 0 . Note that if ν t 0 is the equilibrium state for t 0 z i 1 1 C i , then

The fact that t 0 is a critical point yields

and then for any µ in ≈ y , we get

which concludes the proof.

Let y be such that ≈ y is non-empty. Note that for every µ ∈≈ y , ψ dµ is the constant a 2 q + (a i -a 2 q )y i . For simplicity we denote this quantity by ψ|y . Upper semi-continuity for the entropy yields the existence of measures µ ∈≈ y such that h µ = max{h ν , ν ∈≈ y }.

Then we set h y := h µ .

Lemma 4.5. For every y such that ≈ y is non-empty, set t = ψ|y . Then,

Proof. Let µ t be the DGM associated to the potential βt (a i -a 2 q )1 1 C i . Note the following equality

Set z := (z 1 , . . . , z 2 q -1 ), with z i := βt(a i -a 2 q ).

The properties of the Pressure function (see Subsubsec. 4.2.1) yield that the pressure for βt (a i -a 2 q )1 1 C i is equal to P(βtψ) -βta 2 q = log λ(βt) -βta 2 q .

On the other hand Lemma 4.4 yields (38) h y ≤ P(

It remains to prove that the DGM's are the only maximizers of the quadratic pressure. We define a new functional

This function is defined on some interval M(ψ). If µ is a measure, Lemma 4.5 shows for ψ dµ =: t

This yields that for all t ∈ M(ψ),

Therefore, Lemma 4.6 shows that the maxima for φ r are the maxima for ϕ OS . If a measure µ maximizes the quadratic pressure, then we have and µ must be the DGM µ βt j ψ . This finishes the proof of Theorem 2.


Aurélien Hazan

Volume of the steady-state space of financial flows in a monetary stock-flow-consistent model

Keywords: economics, physics and society, constraint satisfaction, monte-carlo, random network, convex polytope, finance

We show that a steady-state stock-flow consistent macro-economic model can be represented as a Constraint Satisfaction Problem (CSP). The set of solutions is a polytope, which volume depends on the constraints applied and reveals the potential fragility of the economic circuit, with no need to study the dynamics. Several methods to compute the volume are compared, inspired by operations research methods and the analysis of metabolic networks, both exact and approximate. We also introduce a random transaction matrix, and study the particular case of linear flows with respect to money stocks.

Our standpoint is to transpose ideas from the field of metabolic networks where steady-state fluxes have been studied as CSP. These studies were in turn inspired by Von Neumann's growth model of production economies [START_REF] Neumann | A Model of General Economic Equilibrium[END_REF]. The results obtained with metabolic networks were successfully compared to experimental data, as in the Red Blood Cell metabolism or the central metabolism of E.coli [START_REF] Wiback | Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space[END_REF][START_REF] Braunstein | Estimating the size of the solution space of metabolic networks[END_REF][START_REF] Orth | What is flux balance analysis?[END_REF][START_REF] Güell | Mapping high-growth phenotypes in the flux space of microbial metabolism[END_REF]. Such systemscale studies reveal some interesting features of metabolisms, for example the cooperation between pathways. It has been shown also that organisms such as E.coli do not necessarily optimize their metabolic fluxes.

The steady-state equilibrium hypothesis is accepted in the field of metabolic networks because of the separation of timescales between metabolic and genetic regulations [START_REF] Heinrich | Metabolic regulation and mathematical models[END_REF][START_REF] Palsson | Systems biology: properties of reconstructed networks[END_REF]. In economy, the existence of cycles and their corresponding time constants has been the subject of many theories and debates. Recent empirical works are able to identify the timescale at which some specific phenomena operate [START_REF]Wavelet applications in economics and finance[END_REF]. In the case of the model examined in this article, we consider that at the time scale that separates two balance sheets (one year), capital accumulation and output growth are slow and will be considered constant (as noted in [START_REF] Piketty | Le capital au XXIe siècle[END_REF], the global annual per capita growth rate of production is 0.8 % on average on the 1700-2012 time interval).

The expected benefits of applying these methods in macro-economics include the analysis of fragilities, notably the sensitivity to arbitrary flow constraints, such as shortages. Indeed, the volume of the solution space evoked above is immediately impacted when constraints are added or removed, and can reveal the flexibility or rigidity of financial flows subject to perturbations.

In section 1, we detail the model of financial flows that will be used as a benchmark, and present its background from a macro-economic modeling point of view. Comparisons are made with ABM and econophysics. We present the different methods used to compute exactly and approximately the volume of the steady-state solution space. Then in section 2, the experimental results are explained. Finally, sections 3 and 4 are devoted to discussion and conclusion.

Background and methods

Steady-state solution space in a stock-flow-consistent model

In SFC models agents are grouped by sectors (banks, firms, workers, state, central bank) that are linked by money transfers. For example:

• Banks lend money to firms, which pay interests to the former.

• Banks pay interests on deposits made by workers.

• Firms pay wages to workers.

• Workers buy consumption goods to firms.

• Firms invest in capital goods bought from other firms.

Assets and liabilities at a given instant t in time are summarized in a balance sheet, where positive and negative signs stand for uses and sources of money. The balance sheet in Tab. 1 corresponds to the BMW model, discussed in [1, chap. 7], which will be used in this article. In the BMW model, bank issue loans to finance the investments of the productive sector, while households are both consumers and workers. The state and central bank are omitted. Production firms and the banks make no net profit. The net worth of these sectors is zero. The net wealth equals the total tangible capital K.

The transaction matrix sums up all the flows of funds between sectors within a time interval [t, t + ∆t]. Positive and negative signs stand for inflows and outflows of money. They are balanced using a double-entry book-keeping representation where rows sum to zero since each transaction has a counterparty, and columns sum to zero because of the sector's budget constraints.

Tab. 2 shows the transaction matrix corresponding to the BMW model with one agent per sector. After [1, chap. 7], we make the hypothesis that demand terms, with the subscript d, equal supply terms, with the subscript s. Notations are summarized in Tab. 3. In the time-dependent case, the balance sheet at time t + ∆t is obtained adding stocks M, L at time t to changes in deposits and loans ∆M and ∆L that occurred during the time interval [t, t + ∆t].

Households Production Firms Banks

Money deposits +M -M 0 Loans -L +L 0 Fixed capital +K +K Balance (net worth) -V h 0 0 -V h 0 0 0 0
Since in this article the stationary steady-state case is considered, the balance sheet and the transaction matrix are constants, and the change terms ∆M, ∆L are set to zero.

Furthermore in the BMW model a set of behavioral equations expresses several flows as linear functions of the other variables and of fixed parameters. The demand for consumption in equation Eq. (1) depends on the disposable income and on the money deposit of the household:

C d = α 0 + α 1 Y D + α 2 M = α 0 + α 1 (W B s + rM ) + α 2 M (1)
where α 0 , α 1 , α 2 are consumption parameters, and Y D is the disposable income. The depreciation of tangible capital is proportional to its stock:

AF = δK ( 2
)
where δ is the rate of depreciation. Interests are proportional to stocks:

IL s = rL (3)
ID s = rM (4)
where the interest rate r is the same for deposits and loans, for simplicity. The differents constraints (row sums, columns sums, behavioral equations, demand equals supply) can be written in matrix form:

S = {x s.t. ξx = b, ∀i x i ∈ [0, x max i ]} (5)
where x max i sets the maximum flow for each component, and:

x = C s C d I s I d W B s W B d AF s AF d IL s IL d ID s ID d T (6)
and:

ξ =                        
1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 -1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 -1 0 -1 0 -1 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 -1 0 0 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0

                        (7)
Then:

b = 0 0 0 0 0 0 0 0 0 0 -α 0 -(α 1 r + α 2 )M -δK -L rM T (8)
ξ is an m × n matrix with m = 14 and n = 12, m being the number of equations, and n the number of unknown flows. The first six rows of ξ represent the row sums constraints of the transaction matrix. The following four rows represent the budget constraints. The last four rows are equivalent to the behavioral equations. Closed form solutions of S are studied in [1, chap. 7].

Constraints can be added or modified: for example, if a firm goes bankrupt, the loan will not be completely repaid, thus equation IL s = rL may be replaced by IL s ≤ rL. Agents may spend less in consumption than what is prescribed by the behavioral equation, which can be written

C d ≤ α 0 + α 1 Y D + α 2 M .
In section 1.2, the scope of the model is extended to the case of multiple agents per sector. In that case, the linear system will be under-determined. 3: Labels associated with the different monetary variables, after [START_REF] Godley | Monetary economics: an integrated approach to credit, money, income, production and wealth[END_REF]. The subscripts d and s stand for demand and supply.

Many agents per sector

To extend the model of section 1.1 to the case of many agents per sector, the scalars in matrix ξ will be replaced by block matrices. The number of banks, firms and workers are noted nb, nf, nw. Each block will be designed to account not only for the flows and stocks but also for the connectivity between agents.

For example, the first row of matrix ξ, [1, -1, 0, . . . , 0], may be written with blocks instead of scalars, in order to encode the relationship between demand and supply of consumption goods. The positive term, that corresponds to the supply side of consumption, can be replaced by the identity matrix I nf . The negative term, that corresponds to the demand side, can be replaced for example by:

A nf,nh =        0 1 . . . 0 0 0 0 . . . . . . . . . 0 0 0 1 0 . . . 1        , (9)
A nf,nh is an n f × n h matrix, such that A nf,nh [i, j] = 1 if the firm i is selling consumption goods to the household j. In this example, the first and the last households are clients of the same firm.

We suppose that one household can buy consumption goods from one firm only, chosen uniformly at random among the n f firms, and that firms can sell goods to many households, without restriction. Thus the block matrices A nf,nh will be randomly sampled from the set {0, 1} n f ×n h such that columns sum to 1.

Similarly, the following random matrices are introduced:

• B nf,nf encodes investements: firms buy capital goods from one firm only, B nf,nf [i, j] = 1 if the firm j is selling capital goods to the firm i. B nf,nf is randomly sampled from the set {0, 1} n f ×n f with column sums and row sums equal to 1.

• C nf,nh encodes wages: one firm pays wages to many households, households get a wage from one firm only. C nf,nh [i, j] = 1 if firm i pays a wage to household j. C nf,nh is randomly sampled from the set {0, 1} n f ×n h with column sums equal to 1.

• D nb,nf encodes interests on loans: one bank grants loans to many firms which pay interests in return, firms get loans from one bank only, and pay interests to this bank only. D nb,nf [i, j] = 1 if bank i is being paid interests by firm j. D nb,nf is randomly sampled from the set {0, 1} n b ×n f with column sums equal to 1.

• E nb,nh encodes interests on deposits: each household keeps their deposit on one bank account. Banks pay interests in return, and have many accounts opened for their clients. E nb,nh [i, j] = 1 if bank i pays interests to household j. E nb,nh is randomly sampled from the set {0, 1} n b ×n h with column sums equal to 1.

In Tab. 4 we give an example of balance sheet extended to the case of many agents per sector, with random connectivity as explained above. The associated transaction matrix is written in Tab. 5. In the latter, an example of random choice by households of the firm they buy goods from can be observed. The connectivity of flows of interests and changes in loans and deposits respects the one randomly defined in Tab. [START_REF] Kirman | The Economic Crisis is a Crisis for Economic Theory[END_REF].

The following constraint satisfaction problem in matrix form sums up the set of constraints resulting from the balance sheet in Tab. 4, the transaction matrix in Tab. 5, and the behavioral equations in eq.(1-4):

S = {x s.t. ξx = b, ∀i x i ∈ [0, x max i ]} (10)
where:

ξ =                I nf -A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -B -I nf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C -I nf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I nf -I nf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I nb -D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I nb -E 0 0 0 0 -I nh 0 0 I nh 0 0 0 0 0 I nh 0 0 0 0 I nf 0 I nf 0 0 -I nf 0 -I nf 0 -I nf 0 0 0 0 0 0 0 0 -I nf 0 0 I nf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I nb 0 0 -I nb 0 0 0 0 -I nh 0 0 α 1 I nh 0 0 0 0 0 0 0 -(α 1 r + α 2 )I nh 0 0 0 0 0 0 0 0 0 I nf 0 0 0 0 0 0 -δI nf 0 0 0 0 0 0 0 0 0 I nf 0 0 0 -rI nf 0 0 0 0 0 0 0 0 0 0 0 I nh 0 -rI nh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 . . . 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 . . . 1 0 0 0 0 0 0 0 0 0 0 0 0 0 I nf -I nf 0 0 0 0 0 0 0 0 0 0 0 0 -E D 0                (11)

Households

Firms Banks Interests and changes in loans and deposits are set according to the balance sheet. In the stationay case, ∆L = ∆M = 0.

1 2 3 1 2 1 2 Money deposits M 1 -M 1 0 M 2 -M 2 0 M 3 -M 3 0 Loans -L 1 L 1 0 -L 2 L 2 0 Fixed capital K 1 K 1 K 2 K 2 Balance (net worth) -V h1 -V h2 -V h3 -i V hi 0 0 0 0 0 0 0 0

Households Production Firms Banks

Current Capital Current Capital 1 2 3 1 2 1 2 1 2 1 2 Consumption -C d1 C d1 -C d2 C d2 -C d3 C d3 Investment I s1 -I s1 I s2 -I s2 Wage W B s1 -W B s1 W B s2 -W B s2 W B s3 -W B s3 Depreciation -AF 1 AF 1 -AF 2 AF 2 Interest on loans -IL 1 IL 1 -IL 2 IL 2 Interest on deposits ID 1 -ID 1 ID 2 -ID 2 ID 3 -ID 3 Change in loans ∆L 1 -∆L 1 ∆L 2 -∆L 2 Change in deposits -∆M 1 ∆M 1 -∆M 2 ∆M 2 -∆M 3 ∆M 3 0 0 0 0 0 0 0 0 0 0 0
Rows 1 to 6 of ξ express the row sums constraints of the transaction matrix. Rows 7 to 10 represent the column sums. Rows 11 to 14 are equivalent to the behavioral equations. The last rows correspond to the balance sheet.

Compared to eq. ( 6), x now includes the individual stocks M i , L j , K k :

x = C T s C T d I T s I T d W B T s W B T d AF T s AF T d IL T s IL T d ID T s ID T d M T L T K T T (12)
where x is written as a concatenation of vectors. As an example, M T = [M 1 , . . . , M nh ] T . We also write:

b = 0 nf 0 nf 0 nf 0 nf 0 nb 0 nb 0 nh 0 nf 0 nf 0 nb α 0 .1 nh 0 nf 0 nf 0 nh M tot M tot 0 nf 0 nb T (13)
where 0 n and 1 n are vectors, and M tot is the total quantity of deposits.

The dimension of the nullspace of ξ is then determined numerically for a specific set of parameters consistent with [START_REF] Godley | Monetary economics: an integrated approach to credit, money, income, production and wealth[END_REF], and such that nb < nf < nw = 100. We find that in the cases examined in Fig. 1, the matrix ξ doesn't have full rank, and that consequently the system ξx = b is under-determined. The upper bounds x max i are set independently for each variable, and are logic consequences of the positivity of the flows, and of the conservation of the total stock of money :

• ∀i ∈ [1, nh], M i ∈ [0, M tot ]. • ∀i ∈ [1, nf ], L i ∈ [0, M tot ]. • ∀i ∈ [1, nf ], K i ∈ [0, M tot ]. • ∀i ∈ [1, nf ], AF i ∈ [0, δM tot ]. • ∀i ∈ [1, nf ], I i ∈ [0, δM tot ]. • ∀i ∈ [1, nh], ID i ∈ [0, rM tot ]. • ∀i ∈ [1, nf ], IL i ∈ [0, rM tot ].
A graphical representation of monetary transactions with randomly sampled connections between agents is given in Fig. 2.

Because A nf,nh , B nf,nf , C nf,nh , D nb,nf , E nb,nh are realizations of random variables, ξ belongs to an ensemble of random matrices that will be noted Ξ(nb, nf, nh).

When ξ is constant, S defines a convex bounded polytope, that appears in many disciplines, as will be discussed in section 1.3. The solutions to this class of under-specified problems will be examined in sections 1.4 and 1.5. The reader interested in the study of other explicit specifications of flows may look at [START_REF] Keen | Solving the Paradox of Monetary Profits, Economics: The Open-Access[END_REF] and [22]. Lavoie and Godley examine many increasingly detailed models (open economy,...), as well as dynamic specifications of the flows2 .

To the best of our knowledge, the properties of random matrices that correspond to SFC models have not been established in the economics litterature. This representation calls for a closer examination following many works in the field of complex networks studies.

Related works in economics, econophysics, and network science

Double-entry book-keeping is used to establish National Accounts as a means to record estimated flows and stocks consistently, for each country, at several levels of aggregation. National accountants build balance sheets, income and product accounts, in order to compare economies, and to analyze the behavior of economies in time, such as growth. Such accounts are also used in macroeconomic modeling by policymakers to calibrate Computable General Equilibirum models. However, models of this type (e.g. DSGE) do not explicitly enforce stock-flow consistency.

Long before the break of the subprime crisis, many works have established ABM as an alternative, in a move to get rid of hypotheses perceived as unjustified. Equilibrium, rationality, and the hypothesis of the representative agent have been criticized by many economists [START_REF] Kirman | Complex economics: individual and collective rationality[END_REF][START_REF] Fagiolo | Macroeconomic Policy in DSGE and Agent-Based Models, EconomiX Working Papers[END_REF].

Even though all ABM do not verify stock-flow consistency, it is an hypothesis used by several authors [START_REF] Dawid | Agent-based models for economic policy design: Introduction to the special issue[END_REF][START_REF] Dawid | The Eurace@ Unibi Model: An Agent-Based Macroeconomic Model for Economic Policy Analysis[END_REF][START_REF] Seppecher | Flexibility of Wages and Macroeconomic Instability in an Agent-Based Computational Model with Endogenous Money[END_REF][START_REF] Kinsella | Income Distribution in a Stock-Flow Consistent Model with Education and Technological Change[END_REF][START_REF] Bezemer | Causes of Financial Instability: Don't Forget Finance[END_REF].

ABM are praised for their flexibility, their ability to study large populations of heterogeneous and learning agents that interact in possibly non-linear ways. However, ABM need computer intensive simulations and face the problem of being over-parametrized. Calibration is thus difficult and unstable, all the more since empirical studies and reliable data are not abundant compared to the dimension of the parameter space. Recent works tackle the issue of an efficient exploration of the parameter space [START_REF] Salle | Efficient Sampling and Meta-Modeling for Computational Economic Models[END_REF].The issue of comparing a model to experimental data is another problem posed to practitioners, that has been addressed by different methods [START_REF] Guerini | A Method for Agent-Based Models Validation[END_REF]. Under simplifying assumptions such as the aggregation of a subset of agents, theoretical results concerning some macro-economic ABM were recently obtained in a stock-flow-consistent framework, and phase diagrams established, for specific dynamic rules [START_REF] Gualdi | Tipping points in macroeconomic agent-based models[END_REF][START_REF] Gualdi | Monetary Policy and Dark Corners in a stylized Agent-Based Model[END_REF].

The importance of the organization of interactions, embodied by networks, has been stressed in economics [START_REF] Jackson | Social and economic networks[END_REF]. Theoretical works have studied their generic properties (supply chain [START_REF] Weisbuch | From production networks to geographical economics[END_REF], interbank network [START_REF] Delpini | Evolution of Controllability in Interbank Networks[END_REF], trade credit [START_REF] Battiston | Credit chains and bankruptcy propagation in production networks[END_REF]). Empirical studies have laid emphasis upon the topology of real economic networks, such as goods market, national inter-firm trading [START_REF] Watanabe | Biased diffusion on the Japanese inter-firm trading network: estimation of sales from the network structure[END_REF], world trade [START_REF] Garlaschelli | Fitness-Dependent Topological Properties of the World Trade Web[END_REF], global corporate control among transnational corporations [START_REF] Vitali | The Network of Global Corporate Control[END_REF], firm ownership networks. They must sometime be reconstructed, starting from limited information (of equity investments in the stock market [START_REF] Garlaschelli | The scale-free topology of market investments[END_REF], the interbank market [START_REF] Masi | Fitness model for the Italian interbank money market[END_REF]). At the level of individuals, detailed topological information about banking, employment, or consumption seems to be lacking.

Such empirical and theoretical material may serve as an input to shape the set of equations and inequalities discussed above.

Constraint satisfaction problem

As shown in section 1.1, the steady-state solution space associated with eq.( 5) is a bounded convex polytope. Without loss of generality, S can be supposed to have full row rank. The polytope is a n -m dimensional object embedded in an n-dimensional space. Properties of convex polytopes such as their different representations are well studied [START_REF] Avis | lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm[END_REF]. Computing their volume exactly can be achieved by solving the vertex enumeration problem, which is #P -hard. Existing implementations, such as lrs by Avis et al., allow to solve it in reasonable time when n -m is equal to 10 or below. Exact computation methods are employed in linear programming and operations research to solve classical constraint satisfaction problems such as the map coloring problem, and have real-life applications, for example in resource allocation problems. More solutions can be obtained when relaxing hard constraints.

Approximate methods to determine the solution space were proposed by researchers studying metabolic networks and the metabolic steady-state flux space. These methods allow to sample the solution space, to estimate its volume, and to approximate probabilistic properties of the solutions such as the marginal densities [START_REF] Wiback | Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space[END_REF][START_REF] Braunstein | Estimating the size of the solution space of metabolic networks[END_REF][START_REF] Bianconi | On the flexibility of complex systems[END_REF][START_REF] Bianconi | Viable flux distribution in metabolic networks[END_REF][START_REF] Bianconi | Flux distribution of metabolic networks close to optimal biomass production[END_REF][START_REF] Massucci | A Novel Methodology to Estimate Metabolic Flux Distributions in Constraint-Based Models[END_REF]. They can be used to evaluate the sensitivity of the solution space to new constraints. A restriction of this problem known as Flux-Balance-Analysis (FBA) consists in maximizing some objective constraint, which reduces the solution space to a finite set of points [START_REF] Orth | What is flux balance analysis?[END_REF][START_REF] Bianconi | Flux distribution of metabolic networks close to optimal biomass production[END_REF] or to an hyperface. A parallel may be mentioned with the field of random Constraint Satisfaction Problem (rCSP) that stands at the interface between theoretical computer science and statistical physics, and studies sets of solutions to a large number of random constraints, in a boolean space. Many important results such as phase transitions were developed [START_REF] Mezard | Information, physics, and computation, Oxford graduate texts[END_REF]. Although our main focus is the continuous domain, we can take advantage of this theory.

Monte-Carlo sampling of the steady-state solution space

In section 1.4 we mentioned the exact computation of the volume of the steady-state solution space using vertex enumeration, when n -m is small. The result obtained is numeric, and not an analytic expression depending on the parameters of the problem in eq. ( 5).

Another approach proposed in [START_REF] Wiback | Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space[END_REF][START_REF] Almaas | Global organization of metabolic fluxes in the bacterium Escherichia coli[END_REF], suited for larger problems, is Monte-Carlo sampling in the solution space, using a hit-and-run algorithm. The latter needs an initial point inside the convex polytope, which can be found with a relaxation algorithm such as MinOver [START_REF] Krauth | Learning algorithms with optimal stability in neural networks[END_REF]. Then, sampling from a hypersphere, a direction is selected at random. The half-line defined by the starting point and this direction intersects the boundary in a point. This intersection and the starting point form a segment that can be uniformly sampled to get the next point. The procedure defines a Markov Chain that converges to the uniform distribution over the polytope [START_REF] Smith | Efficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions[END_REF][START_REF] Lovasz | Hit-and-Run Mixes Fast[END_REF], in nondeterministic polynomial time O * (n 3 ) after appropriate preprocessing. The notation O * () means that there are logarithmic factors that multiply n 3 , and constants, but are neglected [START_REF] Simonovits | How to compute the volume in high dimension?[END_REF].

The hit-and-run method provides an estimate of the marginal probability density functions (pdf) denoted P i (x) for each unknown i = 1, . . . , N . Correlations between variables can also be estimated, as well as other quantities that can be approximated with a finite sample of the solution space. They characterise the shape of the polytope and are given by:

P i (x) = V ol(S i (x))/V ol(S), S i (x) = {x ∈ S s.t. x i = x} (14)
They can also be written as an integral over all stocks and flows. As remarked by [START_REF] Wiback | Monte Carlo sampling can be used to determine the size and shape of the steady-state flux space[END_REF], the hit-and-run method will not permit us to estimate the absolute volume. Instead, approximations for relative volumes can be obtained, such as r = V ol(S 1 )/V ol(S), where S is the same as in eq.( 5) while S 1 has additional constraints, such as a lower bound for the variable i.

To sum up, using efficient Monte-Carlo sampling methods such as hit-and-run, we will be able to sample medium-sized problems (up to hundreds of variables), to approximate the pdfs, to compute relative volumes. However, we will get no analytic expression of theses quantities, nor approximate entropy. Furthermore, the mixing of the Markov Chain should be examined to ensure convergence. In section 2 we use the implementation by Tervonen et al. [START_REF] Tervonen | Hit-And-Run enables efficient weight generation for simulation-based multiple criteria decision analysis[END_REF].

As evoked in section 1.4, researchers have also used the replica method [START_REF] Bianconi | Viable flux distribution in metabolic networks[END_REF], and message passing algorithms [START_REF] Braunstein | Estimating the size of the solution space of metabolic networks[END_REF][START_REF] Font-Clos | A weighted belief-propagation algorithm for estimating volume-related properties of random polytopes[END_REF][START_REF] Bianconi | Flux distribution of metabolic networks close to optimal biomass production[END_REF] to deal with the problem of estimating marginal densities. The computing time of message passing algorithms scales as O(n) when the factor graph that represents the constraints contains no loop, but without this hypothesis convergence is not guaranteed.

Results

The marginal histograms of the variables of the constraint satisfaction problem defined in section 1.2, where each sector is composed of many agents, are represented in Fig. 3. The solutions were sampled as explained in section 1.5, with ξ constant. The histogram are computed over the solutions, selecting one agent in each sector. The effect of averaging over Ξ(nb, nf, nh) will be examined in section 2.2. The total quantity of deposits is constant, and the parameters have values summarized in Tab. 6.

We can first remark that the pdfs in Fig. 3 can be grouped by shape: the pdf of C d and W B s that both appear in eq.( 1) have an exponential shape. The distribution of ID s has the same shape as that of M . This is consistent with the linear relation in eq.( 4).

The distribution of M is well fitted by a continuous exponential distribution as shown in Fig. 4(a). This can be compared with the empirical finding that, for many industrialized countries, the lower part of the distribution of the wealth can be approximated by an exponential or gamma distribution [51, 2.3]. The influence of re-sampling ξ on this result will be examined in section 2.2.

The pdf of the stock of capital K has the same shape as I d , AF s , AF d which are all related by linear equations. The pdfs of L and IL d have similar shapes, consistently with eq.( 3). This last shape will be discussed in section 2.2.

A comparison between supply-side and demand-side can be made. The mean flow of consumption supply is greater than the mean flow of consumption demand, in agreement with the fact that the number of firms is smaller than the number of households. Similarly, the mean wage supply W B s , which goes to households, is lower than the mean flow W B d paid by firms. The interest on deposit ID s paid to households has a lower mean value than the ID d paid by banks, since the number of banks is smaller than the number of households.

Because the supply C s is for each firm the sum of the demands of their clients, by a central limit argument we can expect C s to converge to a normal law when the number of clients grows.

Because of topological effects, the distributions of flows across agents are heterogeneous, as examplified by the demand for consumption of two agents in Fig. 4(b).

The capital to output ratio γ = K i /Y i = K i /(C s,i + I s,i ) can be computed and its histograms over all firms and all solutions to the CSPs is shown by Fig. 4(c). Econometric studies at the aggregate level report that the value of this ratio has remained between 2 and 8 for many countries, on the long run (see [START_REF] Piketty | Capital is Back: Wealth-Income Ratios in Rich Countries 1700-2010[END_REF] and the corresponding supplementary material3 ).

More generally, these remarks show that the model depicted in this article has interesting properties, can be easily modified, but needs more work before being able to reproduce stylized facts. We discuss this topic in section 3.

Cs

The influence of knock-outs on systems with many agents

The method of relative volumes devised in section 1.5 is used to study the impact of an external perturbation on the volume of the solution space, for example when a flow is constrained. This is termed knock-out in metabolic network analysis, but in the present context can be interpreted as a limitation imposed on monetary variables, such as a credit shortage. Let us define:

S I = {Xs.t ξX = 0, 0 ≤ x i ≤ x max i , ∀i ∈ I} (15)
S α I\I0 = {Xs.t ξX = 0, 0 ≤ x i ≤ x max i , ∀i ∈ I \ I 0 , 0 ≤ x i ≤ αx max i , ∀i ∈ I 0 } (16)
where X = [x 1 , . . . , x n ], n is the number of columns of ξ, I is the set indexing the space of variables, and I 0 the set of variables that will be partially knocked-out, by an amount α. Let us also define:

r V = V ol(S α I\I0 )/V ol(S I ) (17)
The volume ratio r V measures the impact of selective variable knock-out on the solution space. We sample one random topology, as shown in Fig. 2(b), with 2 banks, 3 firms and 10 households. Keeping this topology fixed for the rest of this section, we sample the solution space and summarize the results in Fig. 5, where superposed curves have been removed. All ratios are functions of 1 -α.

It can be remarked in Fig. 5(a) that single flow knock-outs are able to reduce significantly the volume of the steady-state solution space. For example, a 33% knock-out on investement made by one firm (I s ) out of the three defined in this simple model entails a cut by 20% of the volume ratio r V .

The variables most influenced by group knock-outs in Fig. 5(b) appear to be the interest on loans received by banks (IL s ), the investment of firms (I S ), the payment of wages to workers by firms (W B d ). This ranking is partially conserved at the individual and group level, as shown by Fig. 5(a,b).

The variable least influenced by group knock-outs is the demand for consumption goods (C d ). The variables with significant marginal probability on the right of [0, x max i ] (such as IL s in Fig. 3) undergo a large reduction because the tail is cut off. Conversely, variables with a small tail on the right of [0, x max i ], such as C d , show little reduction.

Comparing the left and right panels of Fig. 5, we remark that the influence of group knock-out on volume ratio is larger than single knock-out. This remark is left for further theoretical analysis.

Random matrix

So far, the matrix ξ was considered constant. This hypothesis is interesting when a particular economic network is examined. However, when the emphasis is put more on the properties of a random ensemble of networks than on a particular instance, ξ must be chosen randomly in Ξ(nb, nf, nh). Consequently, in this section, we consider empirical averages over both the solution space and the set of random matrices Ξ(nb, nf, nh).

For computational reasons, the network size is set to nb = 2, nf = 3, nh = 6. Even in this particular case, the set of possible topologies is large, due to its combinatorial nature, but can be sampled exhaustively in reasonable time. Nevertheless, since for each randomly sampled matrix ξ the corresponding solution space S must also be sampled (which scales as O * (n 3 ), as discussed in section 1.5), the number of sampled matrices will be limited to one hundred.

As illustrated by Fig. 6, the main observations made at the beginning of section 2 are recovered: the marginal distributions are grouped in the same way, the difference between mean supply and demand is conserved, and the money deposit M still has an continuous exponential shape.

We expect some effects related to topology to be averaged out, such as the heterogeneity between agents, that was examplified in Fig. 4(b). However, to check this with the requested statistical significance, we need to increase the number of sampled matrices, with an acceptable computational load, as will we discussed in section 3. volume ratio q q q q q q q q q q Cs Cd Is WBd ILs ILd volume ratio q q q q q q q q q q Cs Cd Is WBd ILs (b)

Discussion

The first point we want to emphasize is that both the model and results in the sections above are preliminary. We do not claim that they can be of any use at the moment regarding economic analysis. More work is needed, in collaboration with economists, to improve their design, to evaluate the empirical stylized facts they can reproduce, and eventually to validate the model [START_REF] Guerini | A Method for Agent-Based Models Validation[END_REF] using empirical data. In our view, the capacity to account for economic crises can be seen through the rapid decay of the volume ratio in response to modest fluctuations of the parameters, such as an increase in the knock-out factor. This can be seen independently of dynamical considerations.

In section 1.5 we presented an algorithm to compute the volume of the solution space that belongs to the class of Monte-Carlo methods, and illustrates the uniform sampling property of the hit-and-run strategy. We didn't discuss convergence issues here, but they will become important as the size of the problem increases. Furthermore, alternative estimation methods exist, for example belief propagation that scales almost linearly in system size, and can be applied when the variables are continuous [START_REF] Font-Clos | A weighted belief-propagation algorithm for estimating volume-related properties of random polytopes[END_REF][START_REF] Fernandez-De Cossio-Diaz | Fast inference of ill-posed problems within a convex space[END_REF].

None of these algorithms provide litteral expressions that could be compared to results presented in section 2, such as the expression of absolute or relative volumes. For example, the magnitude of volume reduction provoked by knock-outs is an important quantity from a system-scale point of view. Interestingly, some theoretical results have been developped in the statistical physics literature evoked earlier, and should be compared to our numerical experiments, in future works.

Concerning the complexity of the financial SFC model depicted in section 1.1, we started with a very simple setting, with a limited number of transaction types, one agent per class, no state nor central bank. Flows were constrained in magnitude and depended on stocks. Then, the number of agents increased in section 1.2 which gave us a hint of the influence of topology. The accuracy of the model could be increased if random connectivity matrices were sampled in a random ensemble that corresponds to connectivity patterns observed empiricaly. Seeking inspiration in macro-economic literature (e.g. Lavoie and Godley), we may also add financial constraints on debt ratios at various levels. Transactions not covered in this article can also be added, related to bonds, investments, in an open economy. A major improvement would be to couple the financial side to a production model of the economy, in order to compute prices, demand, unemployment and profit [START_REF] Bardoscia | Complexity driven collapse of economic equilibria[END_REF], but many difficulties can be expected in that direction because of nonlinear relations that transform the linear CSP into a nonlinear one.

The issue of employment can't be addressed directly in this model because all households get a wage from some firm. Unemployed households should be disconnected from firms, and included into another economic circuit, but this is not a feature of the BMW model, nor of the extension proposed here. What can still be studied thanks to the random nature of income in this framework, is the proportion of households whose income is below a given threshold, which could be related, for example, to aggregate demand.

In section 2, following [1, chap. 7], with the hypothesis that flows are linear functions of stocks, we obtained interesting marginal quantities, such as the capital to output ratio. But this raises the issue of choosing the right parameter values. Various strategies can be implemented to address it, such as setting the parameters according to empirical data taken from public statistics, or efficiently exploring the space of parameters [START_REF] Salle | Efficient Sampling and Meta-Modeling for Computational Economic Models[END_REF]. Another point of view is to consider the linear flows parameters as variables defined in specific intervals, and to include them in the sampling scheme used above. Although flows are linear functions of stocks, this problem is also a nonlinear CSP, harder to deal with than a linear one.

On the computational complexity side, a comparison should be made with other classical approaches such as DSGE models and ABM with respect to the number of agents, the sparseness of the network topology, the number of regions or countries, the richness of the financial mechanisms involved. We can remark that in the case of hit-and-run sampling, the main cost is polynomial in the system size during sampling. Then in order to compute all the ratios discussed above, it is not necessary to resample: basic thresholding is sufficient, and is linear in system size.

Furthermore, as reported in the random network community [START_REF] Coolen | Constrained Markovian Dynamics of Random Graphs[END_REF][START_REF] Squartini | Unbiased sampling of network ensembles[END_REF], sampling network matrices ξ with hard topological constraints raises the issue of bias and efficiency, and will have to be controlled for in future works.

Conclusion

We proposed an original strategy to compute macro-economic stocks and flows in a financial economy, inspired by stock-flow consistent models and methods developed in the field of metabolic networks. We show that this approach can be efficiently transposed thanks to approximate Monte-Carlo algorithms designed to solve Constraint Satisfaction Problems. The steady-state in variable space is seen as a polytope included in a large dimensional space, which weighs equally all configurations of the financial stocks and flows that are consistent with the constraints enforced by double-entry accounting.

We proposed a random connectivity extension of the linear flow model BMW by [START_REF] Godley | Monetary economics: an integrated approach to credit, money, income, production and wealth[END_REF], and have obtained a numerical approximation of the probability density of stocks, flows, and the capital to output ratio. The money stock of households was found to be exponential in its lower part, which is reminiscent of standard works in econophysics [START_REF] Chakrabarti | Econophysics of income and wealth distributions[END_REF].

Flow knock-outs can be used to model economic phenomena such as credit shortages. We show that different types of constraints have distinct effects on the volume of the solution space, that can be interpreted as characterizing the flexibility of the financial flows. Rapid decay of the volume ratio in response to modest fluctuations of the parameters can be interpreted as crises, independently of dynamical specifications.

Inside the class of SFC models, our approach fills a gap between SFC-ABM on one hand, and aggregate SFC models on the other hand, because the system size can scale up to thousands of agents while preserving the possibility of a theoretical analysis, and heterogeneity among agents. This comes at the cost of a simplification of the model, notably the hypothesis of non-equilibrium stationary state, with linear flows. We discussed many potential improvements concerning the algorithms, the complexity of the model, and the relation to empirical data, and will deal with it in future works.

Appendix A. Acknowledgements

The author wishes to thank two anonymous reviewers for their helpful comments.

Open-source software were used to perform this research: Python, R, the R hit-and-run package [START_REF] Tervonen | Hit-And-Run enables efficient weight generation for simulation-based multiple criteria decision analysis[END_REF], lrs, graphviz, pygraphviz.

Figure 1 :

 1 Figure 1: Dimension of the nullspace of ξ determined numerically with r = 0.03, α 1 = 0.75, α 2 = 0.1, δ = 0.1, and nw = 100. This dimension is computed only for values of nf identified by the marker •. nb, nf, nw are such that nb < nf < nw

Figure 2 :

 2 Figure 2: Graph of a subset of monetary transactions. cur,cap,w,b stand for firm's current and capital accounts, worker's deposit accounts, bank's deposit accounts. (a) single agent in each class; (b) random connectivity, with 2 banks, 3 firms, 4 workers.

Figure 3 :Table 6 :Figure 4 :

 364 Figure 3: Histograms corresponding to marginal pdf of individual steady-state monetary stocks and flows labelled according to Tab.3, with n b = 2, n f = 3, n h = 10. The number of samples solution of the constraint satisfaction problem is N = 3.10 5 . Subscripts have been dropped for clarity.

Figure 5 :

 5 Figure 5: Impact of knock-out on volume ratios, in the case of a single draw of the random connectivity model depicted in sec.1.1, with 2 banks, 3 firms and 10 workers. Volume ratio appear as functions of (1 -α). (left) single flow knock-out; (right) group flow knock-out. Flow labels are explained in Tab. 3. The number of sampled points is n = 3.10 5 .

Table 1 :

 1 Balance sheet of the BMW model. M, L, K are the money deposits, loans, and tangible capital. V h is the net worth of households.

		Households Production Firms		Banks
			Current Capital Current Capital
	Consumption	-C d	C s				0
	Investment		I s	-I d			0
	Wages	W B s	-W B d				0
	Depreciation		-AF d	AF s			0
	Interest on loans		-IL d		IL s		0
	Interest on deposits ID s			-ID d		0
	Change in loans			∆L		-∆L	0
	Change in deposits -∆M				∆M	0
		0	0	0	0	0	0

Table 2 :

 2

Transaction matrix of the BMW model. In the stationary case, ∆L = ∆M = 0.

Table

	Variable	Label
	money deposit	M
	capital	K
	loans to firms	L
	investment	I
	interest on loans	IL
	wage bill	W B
	depreciation allowance	AF
	interest on workers deposits ID
	consumption of workers	C

Table 4 :

 4

Example of balance sheet of the BMW model with many agents nw = 3, nf = 2, nb = 2. Households and firms randomly choose their bank. M i , L j , K k are the individual money deposits, loans, and tangible capital. V hi is the net worth of individual households.

Table 5 :

 5 Example of transaction matrix of the BMW model with many agents nw = 3, nf = 2, nb = 2. Households randomly choose firms to buy consumption goods. Firms randomly choose other firms to buy capital goods, and the workers they hire.

see an implementation https://github.com/kennt/monetary-economics

http://piketty.pse.ens.fr/fr/capitalisback

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