Correlation Inequalities and Monotonicity Properties of the Ruelle Operator
Let be the symbolic space endowed with a partial order , where , if , for all . A function is called increasing if any pair , such that , we have A Borel probability measure over is said to satisfy the FKG inequality if for any pair of continuous increasing functions and we have . In the first part of the paper we prove the validity of the FKG inequality on Thermodynamic Formalism setting for a class of eigenmeasures of the dual of the Ruelle operator, including several examples of interest in Statistical Mechanics. In addition to deducing this inequality in cases not covered by classical results about attractive specifications our proof has advantage of to be easily adapted for suitable subshifts. We review (and provide proofs in our setting) some classical results about the long-range Ising model on the lattice and use them to deduce some monotonicity properties of the associated Ruelle operator and their relations with phase transitions.
As is widely known, for some continuous potentials does not exists a positive continuous eigenfunction associated to the spectral radius of the Ruelle operator acting on . Here we employed some ideas related to the involution kernel in order to solve the main eigenvalue problem in a suitable sense for a class of potentials having low regularity. From this we obtain an explicit tight upper bound for the main eigenvalue (consequently for the pressure) of the Ruelle operator associated to Ising models with interaction energy. Extensions of the Ruelle operator to suitable Hilbert Spaces are considered and a theorem solving to the main eigenvalue problem (in a weak sense) is obtained by using the Lions-Lax-Milgram theorem. We generalize results due to P. Hulse on attractive -measures. We also present the graph of the main eigenfunction in some examples - in some cases the numerical approximation shows the evidence of not being continuous.
Monotone functions, Correlation Inequalities, FKG inequality, Ruelle operator,
Eigenfunctions, Eigenprobabilities, Equilibrium states, Measurable Eigenfunctions.
MSC2010: 37D35, 28Dxx, 37C30. \@footnotetextThe authors are supported by CNPq-Brazil.
The primary aim of this paper is to relate the Fortuin-Kasteleyn-Ginibre (FKG) inequality to the study of the main eigenvalue problem for Ruelle operator associated to an attractive potential having low regularity (meaning lives outside of the classical Hölder, Walters and Bowen spaces).
The FKG inequality [FKG71] is a strong correlation inequality and a fundamental tool in Statistical Mechanics. An earlier version of this inequality for product measures was obtained by Harris in [Har60]. Holley in [Hol74] generalized the FKG inequality in the context of finite distributive lattice. In the context of Symbolic Dynamics the FKG inequality can be formulated as follows. Let us consider the symbolic space with an additional structure which is a partial order , where , if , for all . A function is said increasing if for all , such that , we have A Borel probability measure over will be said to satisfy the FKG inequality if for any pair of continuous increasing functions and we have
In Probability Theory such measure are sometimes called positively associated.
Our first result asserts that for any potential (Definition 2) the probability measure defined in (3) satisfies the FKG inequality. As a consequence of this result we are able to shown that at least one eigenprobability of , associated to its spectral radius, must satisfies the FKG inequality. Some similar results for -measures where obtained by P. Hulse in [Hul06, Hul97, Hul91]. Potentials satisfying a condition similar to Definition 2 are called attractive potentials on these papers, which is a terminology originated from attractive specifications sometimes used in Statistical Mechanics.
Establishing FKG inequality for continuous potentials with low regularity is a key step to study, for example, the Dyson model on the lattice , within the framework of Thermodynamic Formalism. A Dyson model (see [Dys69]) is a special long-range ferromagnetic Ising model, commonly defined on the lattice . The Dyson model is a very important example in Statistical Mechanics exhibiting the phase transition phenomenon in one dimension. This model still is a topic of active research and currently it is being studied in both lattices and , see the recent preprints [vEN16, JÖP16] and references therein. In both works whether the DLR-Gibbs measures associated to the Dyson model is a -measures is asked.
In [JÖP16] the authors proved that the Dyson model on the lattice has phase transition. This result is an important contribution to the Theory of Thermodynamic Formalism since very few examples of phase transition on the lattice are known (see [BK93, CL15, Geo11, Hof77, Hul06]). In this work is also proved that the critical temperature of the Dyson model on the lattice is at most four times the critical temperature of Dyson model on the lattice . The authors also conjectured that the critical temperature for both models coincides. We remark that the explicit value of the critical temperature for the Dyson model on both lattices still is an open problem. Moreover there are very few examples in both Thermodynamic Formalism and Statistical Mechanics, where the explicit value of the critical temperature is known. A remarkable example where the critical temperatures is explicitly obtained is the famous work by Lars Onsager [Ons44] and the main idea behind this computation is the Transfer Operator.
Although the Ruelle operator (associated to the potential ) have been intensively studied, since its creation in 1968, and became a key concept in Thermodynamic Formalism a little is known about , when is the Dyson potential. The difficult in using this operator to study the Dyson model is the absence of positive continuous eigenfunctions associated to the spectral radius of its action on . An alternative to overcome this problem is to consider extensions of this operator to larger spaces than , where a weak version of Ruelle-Perron-Frobenius theorem can be obtained. We point out that continuous potentials may not have a continuous positive eigenfunction but the dual of the Ruelle operator always has an eigenprobability. Here we study the extension of the Ruelle operator to the Lebesgue space , where is an eigenmeasure for . We study the existence problem of the main eigenfunction in such spaces by using the involution kernel and subsequently the Lions-Lax-Milgram theorem.
In another direction we show how to use the involution kernel representation of the main eigenfunction and the FKG inequality to obtain non-trivial upper bound for the topological pressure of potentials of the form
which is associated to a long-range Ising model, when is suitable chosen. A particular interesting case occurs when with (see end of section 5 in [CL14] for the relation with the classical Long-range Ising model interaction).
The above mentioned upper bound coincides with the topological pressure of a product-type potential (which is different but similar to the previous one) given by
See [CDLS17] for the computation of the topological pressure of . In some sense we can think of this model as a simplified version of the previous one. In this simpler model is possible to exhibit explicit expressions for the eigenfunction and eigenprobability of the Ruelle operator , see [CDLS17].
Suppose that for all , for both potentials (1) and (2). Although the potentials and have completely different physical interpretations (two-body interactions versus self interaction) from the Thermodynamic Formalism point of view they have interesting similarities. For example, in the simplified model (case (2)) one can show that the Ruelle operator stops having positive continuous eigenfunction if (see [CDLS17]). On the other hand, in a similar fashion, for the potential in case (1) and , Figure 5 on section 8 - obtained via a numerical approximation - seems to indicate that there exists a non-continuous eigenfunction.
When , the eigenfunctions associated to both potentials are very well-behaved and they belong to the Walters space. Although for we do not have phase transition for the potential the unique non-negative eigenfunction for is such that its values oscillate between zero and infinity in any cylinder subset of . On the other hand, if then we know from [JÖP16] that there is phase transition for the potential in the sense of the existence of two eigenprobabilities. These observations suggest that the main eigenfunction of carries information about phase transition for the potential .
In Section 5 we show how to use the involution kernel in order to construct an “eigenfunction” (the quotes is because of they are only defined on a dense subset of ) for the Dyson model associated to the spectral radius of the Ruelle operator.
Some results of P. Hulse are generalized to non normalized potentials in Section 7 and use some stochastic dominations coming from these extensions to obtain uniqueness results for eigenprobabilities for a certain class of potentials with low regularity.
2 Increasing Functions and Correlation Inequalities
Let be the set of the non-negative integers and be any fixed positive number. Consider the symbolic space and the left shift mapping defined for each by . As usual we endow with its standard distance , where , where . As mentioned before we consider the partial order in , where , iff , for all . A function is called increasing (decreasing) if for all such that , we have that (). The set of all continuous increasing and decreasing functions are denoted by and , respectively.
For each , and will be convenient in this section to use the following notations
A function will be called a potential. For each potential , and we define
For any fixed and we define a probability measure over by the following expression
and is the Dirac measure supported on the point . The normalizing factor is called partition function (associated to the potential ).
Let be given. A function is called a differentiable extension of a potential if for all we have and for all the following partial derivatives exist and the mappings
are continuous for any fixed .
To avoid a heavy notation, a differentiable extension of a potential will be simply denoted by . Note that the Ising type potentials are examples of continuous potentials admitting natural differentiable extensions.
Definition 2 (Class potential).
We say that a continuous potential belongs to class if it admits a differentiable extension satisfying: for any fixed , , we have that
is an increasing function from to .
Let be fixed and two real increasing functions, with respect to the partial order , depending only on its first coordinates. The main result of the next section states that for all potential in the class the probability measure given by (3) satisfies the FKG Inequality
for any choice of .
In what follows we exhibit explicit examples of potentials in the class .
2.1 Dyson Potential
An Ising type model, on the lattice , in Statistical Mechanics is a model defined in the symbolic space , with . Here we call a Ising type potential any real function of the form where and are fixed real numbers satisfying . An interesting family of such potentials is given by
When the potential is sometimes called Dyson potential. It is worth to mention that a Dyson potential is not an increasing, decreasing or Hölder function.
As will be shown latter the probability measure determined by the potential given in (6) satisfies the correlation inequality, of the last section, for any choice of .
Remark. The Dyson potential for any fixed and belongs to the class . Indeed, a straightforward computation shows that
for some constant , which depends on and but not on . From this expression the condition (4) can be immediately verified. More generally, any Ising type potential with for all , satisfies the hypothesis of Theorem 1. In this case, the potential is sometimes called ferromagnetic potential.
2.2 The FKG Inequality
The results obtained in this section are inspired in the proof of the FKG inequality for ferromagnetic Ising models presented in [Ell06]. In this reference the inequality is proved under assumptions on the local behavior of the interactions of the Ising model but here our hypothesis are about the global behavior of the potential. For sake of simplicity we assume that . The arguments and results obtained here can be immediately generalized for any other choice of .
In order to keep the paper self-contained we recall the following classical result.
Let and a probability space. If are increasing functions then
Since and are increasing functions, then for any pair we have . By integrating both sides of this inequality, with respect to the product measure , using the elementary properties of the integral and is a probability measure we finish the proof. ∎
Now we present an auxiliary combinatorial lemma that will be used in the proof of Theorem 1.
Let , fixed and a continuous function. Then the following identity holds for all
By using the definitions of and , respectively we get
To shorten the notation in the remaining of this section, we define for each , and the following weights
Let and be fixed, an increasing function, depending only on its first coordinates . If the potential belongs to the class and satisfies the inequality (5), then
is a real increasing function.
We first observe that the integral in (9) is well-defined because admits a differentiable extension defined in whole space .
By using that depends only on its first coordinates we have the following identity for any
Since belongs to the class follows from the expression (8) that has continuous derivative and therefore to prove the lemma is enough to prove that
By using the quotient rule we get that the derivative appearing in the above expression is equal to
Note that the last term in the rhs above is equal to
Let be a potential in the class . For any fixed and for all the probability measure
where is the standard partition function, satisfies the correlation inequality (5).
The proof is by induction in . The inequality (5), for , follows from a straightforward application of Lemma 1. Indeed, for any fixed the mappings and are clearly increasing. By thinking of these maps as functions from to and as a probability measure over , we can apply Lemma 1 to get the conclusion.
The induction hypothesis is formulated as follows. For some assume that for all and any pair of real continuous increasing functions and , depending only on its first coordinates, we have
Now we prove that satisfy (5). From the definition we have that
By using the induction hypothesis on both terms in the rhs above we get that
where the last equality is ensured by the Lemma 2. ∎
2.3 FKG Inequality and the Ising Model
Let and be a collection of real numbers belonging to the set
For each we define a real function by following expression
For each , and we define a probability measure by the following expression
where is the partition function. In the next section we show that for suitable choices of and the expression (17) can be rewritten in terms of the Ruelle operator.
Theorem 2 (FKG-Inequality).
Let , and so that for any pair . If are increasing functions depending only on its first coordinates, then
Note that the Hamiltonian admits a natural differentiable extension to a function defined on and so for any , depending on its first coordinates, the following partial derivatives exist and are continuous functions
Under the hypothesis of Theorem 2 we have
In particular, if then
Under the hypothesis of Theorem 2 if then
By considering the natural differentiable extension of to we can proceed as in (2.2) obtaining
By using that we get from (16) that the mapping is an increasing function. So we can apply the FKG inequality to the rhs above to ensure that function
is coordinate wise increasing and therefore the result follows. ∎
To lighten the notation , when or similarly , we will simply write or , respectively. If the parameters and are clear from the context they will be omitted.
Under the hypothesis of Theorem 2 we have
The proof of these inequalities are similar, so it is enough to present the argument for the first one. From Corollary 1 we get that
By using the definition of we have
A straightforward computation shows that
The expression below is clearly uniformly bounded away from zero and infinity, when goes to infinity
Finally by using l’hospital rule one can see that
Piecing the last four observations together, we have
Under the hypothesis of Theorem 2 we have