---
abstract: 'In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem to compute, in terms of the maximal eigendata of the Ruelle operator, the pressure derivative of translation invariant spin systems taking values on a general compact metric space. On this setting the absence of metastable states for continuous potentials on one-dimensional one-sided lattice is proved. We apply our results, to show that the pressure of an essentially one-dimensional Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is Fréchet differentiable, on a suitable Banach space. Additionally, exponential decay of the two-point function, for this model, is obtained for any positive temperature.'
author:
- 'E. A. Silva'
title: 'Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach'
---

fnsymbolroman

Introduction
============

The Ruelle operator was introduced by David Ruelle in the seminal paper [@Ruelle-1968], in order to prove the existence and uniqueness of the Gibbs measures for some long-range Statistical Mechanics models in the one-dimensional lattice. Ever since the Ruelle operator has become a standard tool in a variety of mathematical fields, for instance, in Dynamical Systems, and other branches of Mathematics and Mathematical Physics. The Ruelle operator was generalized in several directions and its generalizations are commonly called transfer operators. Transfer operators appear in IFS theory, Harmonic Analysis and $C^{*}$-algebras, see for instance [@Excel-Lopes; @Lau; @Straub] respectively.

In Dynamical Systems the existence of Markov Partitions allows one to conjugate uniform hyperbolic maps on compact differentiable manifolds with the shift map in the Bernoulli space. For more details see, for example, [@bo] and references therein.

A field in which the Ruelle operator formalism has also been proved useful is the Multifractal Analysis. Bowen, in the seminal work [@boo], has established a relationship between the Hausdorff dimension of certain fractal sets and topological pressure, for more details see [@boo; @ma] and also the introductory texts [@Barreira; @Pesin].

The classical Thermodynamic Formalism was originally developed in the Bernoulli space $M^{\mathbb{N}}$, with $M$ being a finite alphabet, see [@PP]. The motivation to consider more general alphabets from the dynamical system point of view is given in [@Sarig; @1; @Sarig; @2], where proposed models with infinite alphabet $M=\mathbb{N}$ are used to describe some non-uniformly hyperbolic maps, for instance, the Manneville-Pomeau maps. Unbounded alphabets as general standard Borel spaces, which includes compact and non-compact, are considered in details in [@Geogii88].

In [@le] the authors considered the alphabet $M=S^1$ and a Ruelle operator formalism is developed. Subsequently, in [@LMMS], this formalism was extended to general compact metric alphabets. Those alphabets do not fit in the classical theory, since the number of preimages under the shift map may be not countable. To circumvent this problem the authors considered an a priori measure $\mu$ defined on $M$, and so a generalized Ruelle operator can be defined and a version of the Ruelle-Perron-Frobenius Theorem is proved. In this general setting concepts of entropy and pressure are also introduced. A variational principle is obtained in [@LMMS]. The authors also show that their theory can also be used to recovery some results of Thermodynamic Formalism for countable alphabets, by taking the one-point compactification of $\mathbb{N}$, and choosing a suitable a priori measure $\mu$.

In Classical Statistical Mechanics uncountable alphabets shows up, for example, in the so-called $O(n)$ models with $n\geq 2$. These are models on a $d$-dimensional lattice for which a vector in the $(n-1)$-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. More specifically, let $n\geq 1$ be an integer and let $G=(V(G), E(G))$ a finite graph. A configuration of the [*Spin $O(n)$ model*]{} on $G$ is an assignment $\sigma:V(G)\to S^{n-1}$, we denote by $\Omega:=(S^{n-1})^{V(G)}$ the space of configurations. At inverse temperature $\beta\in (0, \infty),$ configurations are randomly chosen from the probability measure $\mu_{G,n, \beta}$ given by $$d\mu_{G,n, \beta}(\sigma):=\dfrac{1}{Z_{G, n, \beta}}
\exp ( \beta \sum_{u,v \in E(G)} \sigma_u\cdot \sigma_v) \ d\sigma,$$ where $\sigma_u\cdot \sigma_v$ denotes the inner product in $\mathbb{R}^n$ of the assignments $\sigma_u$ and $\sigma_v$, $Z_{G, n, \beta}=\int_{\Omega}
\exp( 
\beta \sum_{u,v \in E(G)} \sigma_u\cdot \sigma_v)\, d\sigma
$ and $d\sigma$ is the uniform probability measure on $\Omega$.

Special cases of these models have names of their own: when $n=0$ this model is the Self Avoiding Walking (SAW); when $n=1$, this model is the Ising model, when $n=2$, the model is called the $XY$ model; finally for $n=3$ this model is called the Heisenberg model, see [@le; @Ellis; @GJ-Book; @Geogii88; @ONModels] for details.

In [@CL-SIL] the authors generalize the previous Ruelle-Perron-Frobenius Theorem for a more general class of potentials, satisfying what the authors called weak and strong Walters conditions, which in turn are natural generalizations for the classical Walters condition. The regularity properties of the pressure functional are studied and its analyticity is proved on the space of the Hölder continuous potentials. An exponential decay rate for correlations are obtained, in the case where the Ruelle operator exhibits the spectral gap property. An example, derived from the long-range Ising model, of a potential in the Walters class where the associated Ruelle operator has no spectral gap is given.

One of the main results of this work provide an explicit expression for the derivative of the pressure $P:C^{\alpha}(\Omega)\to \mathbb{R}$, where $\Omega \equiv M^{\mathbb{N}}$ and $M$ is a general compact metric space. To be more precise, we show that $$\tag{1}\label{pressure-derivative-0}
P^{'}(f)\varphi=\int_{\Omega}\varphi h_f\, d\nu_f,$$ where $h_f$ and $\nu_f$ are eigenfunction and eigenmeasure of the associated Ruelle operator.

The proof follows closely the one given in [@ma], where the expression is obtained in the context of finite alphabet. We also prove the existence of the limit $$P(f)=\lim_{n\to \infty}\dfrac{1}{n}\log \mathscr{L}_{f}^n{\bf 1}(x)$$ in the uniform sense, for any continuous potential $f$. We would like to point out that the existence of this limit in this setting has been proved in [@CL16]. We give a new proof of this fact here for two reasons: first, it is different from the proof presented in [@CL16] and we believe it is more flexible to be adapted to other contexts; second, some pieces of it are used to compute the pressure derivative.

In the last section we apply our results in cases where $M=(S^2)^{\mathbb{Z}}$ endowed with a suitable a priori DLR-Gibbs measure. We introduce a Heisenberg-type model on the lattice $\mathbb{N}\times \mathbb{Z}$, depending on a real parameter $\alpha$, and use the Ruelle operator to obtain differentiability of the pressure and exponential decay rate for the two-point function for any positive temperature.

Basic Definitions and Results
=============================

In this section we set up the notation and present some preliminaries results. Let $M=(M,d)$ be a compact metric space, equipped with a Borel probability measure $\mu:\mathscr{B}(M)\to [0,1]$, having the whole space $M$ as its support. In this paper, the set of positive integers is denoted by $\mathbb{N}$. We shall denote by $\Omega=M^{\mathbb{N}}$ the set of all sequences $x=(x_1,x_2,\ldots )$, where $x_i \in M$, for all $i\in {\mathbb N}$. We denote the left shift mapping by $\sigma:\Omega\to\Omega,$ which is given by $\sigma(x_1,x_2,\ldots)=(x_2,x_3,\ldots)$. We consider the metric $d_{\Omega}$ on $\Omega$ given by $$d_{\Omega} (x,y)
=
\sum_{n=1}^{\infty} \frac{1}{2^n}d(x_n,y_n).$$ The metric $d_{\Omega}$ induces the product topology and therefore it follows from Tychonoff’s theorem that $(\Omega,d_{\Omega})$ is a compact metric space.

The space of all continuous real functions $C(\Omega, \mathbb R)$ is denoted simply by $C(\Omega)$ and will be endowed with the norm $\|\cdot\|_0$ defined by $\|f\|_0=\sup_{x\in \Omega} |f(x)|,$ which in turn is a Banach space. For any fixed $0< \alpha< 1$ we denote by $C^{\alpha}(\Omega)$ the space of all $\alpha$-Hölder continuous functions, that is, the set of all functions $f:\Omega\to\mathbb{R}$ satisfying $$\mathrm{Hol}_{\alpha}(f)
=
\sup_{x,y\in\Omega: x\neq y}
\dfrac{|f(x)-f(y)|}{d_{\Omega}(x,y)^{\alpha}}
<+\infty.$$

We equip $C^{\alpha}(\Omega),~0< \alpha< 1,$ with the norm given by $\|f\|_{\alpha}= \|f\|_0+ \mathrm{Hol}(f)$. We recall that $(C^{\alpha}(\Omega),\|\cdot\|_{\alpha}) $ is a Banach space for any $0< \alpha< 1$.

Our *potentials* will be elements of $C(\Omega)$ and, in order to have a well defined Ruelle operator when $(M,d)$ is a general compact metric space, we need to consider an [*a priori measure*]{} which is simply a Borel probability measure $\mu:\mathscr{B}(M)\to \mathbb{R}$, where $\mathscr{B}(M)$ denotes the Borel $\sigma$-algebra of $M$. For many of the most popular choices of an uncountable space $M$, there is a natural a priori measure $\mu$. Throughout this paper the a priori measure $\mu$ is supposed to have the whole space $M$ as its support. The Ruelle operator $\mathscr{L}_f: C(\Omega) \to C(\Omega)$ is the mapping sending $\varphi$ to $\mathscr{L}_{f}(\varphi)$ defined for any $x\in\Omega$ by the expression $$\mathscr{L}_f(\varphi)(x)
=
\int_M e^{f(ax)}\varphi(ax)d\mu(a),$$ where $ax$ denotes the sequence $ax=(a, x_1, x_2, \ldots)\in \Omega$. The classical Ruelle operator can be recovered on this setting by considering $M=\{0,1,\ldots,n\}$ and the a priori measure $\mu$ as the normalized counting measure.

\[teo-RPF-compacto\] Let $(M,d)$ be a compact metric space, $\mu$ a Borel probability measure on $M$ having full support and $f$ a potential in $C^{\alpha}(\Omega)$, where $0<\alpha<1$. Then $\mathscr{L}_f: C^{\alpha}(\Omega) \to C^{\alpha}(\Omega)$ has a simple positive eigenvalue of maximal modulus $\lambda_f$, and there are a strictly positive function $h_f$ and a Borel probability measure $\nu_{f}$ on $\Omega$ such that,

-   $\mathscr{L}_f h_f=\lambda_f h_f,$ $\mathscr{L}^{*}_f\nu_f=\lambda_f\nu_f$;

-   the remainder of the spectrum of $\mathscr{L}_f: C^{\alpha}(\Omega) \to C^{\alpha}(\Omega)$ is contained in a disc of radius strictly smaller than $\lambda_f$;

-   for all continuous functions $\varphi\in C(\Omega)$ we have $$\lim_{n\to\infty}
    \left\|
    \lambda_{f}^{-n}\mathscr{L}^{n}_{f}\varphi-h_f\int \varphi\,  d\nu_{f}
    \right\|_0
    =
    0.$$

See [@le] for the case $M=S^1$ and [@LMMS] for a general compact metric space.

Following the references [@le; @LMMS] we define the entropy of a shift invariant measure and the pressure of the potential $f$, respectively, as follows $$h(\nu)
=
\inf_{f \in C^{\alpha}(\Omega)}
\left\{-\int_{\Omega}f d\nu+ \log\lambda_{f}  \right\}
\; \text{and} \;\,
P(f)
=
\sup_{\nu\in \mathscr{M}_{\sigma}}
\left\{h(\nu)+\int_{\Omega}f\, d\nu \right \},$$ where $\mathscr{M}_{\sigma}$ is the set of all shift invariant Borel probability measures.

\[Principio variacional\] For each $f \in C^{\alpha}(\Omega)$ we have for all $x\in \Omega$ that $$P(f)
=
\lim_{n\to\infty}
\frac{1}{n}
\log[ \mathscr{L}_{f}^n({\bf 1})(x)]
=
\log\lambda_{f}
=\sup_{\nu\in \mathscr{M}_{\sigma}}
\left\{h(\nu)+\int_{\Omega}f\, d\nu \right \}.$$ Moreover the supremum is attained by $m_f=h_f \nu_f$.

See [@LMMS] Corollary 1.

\[Pressure Analiticity\] The function defined by $C^{\alpha}(\Omega)\ni f\mapsto P(f)\in \mathbb{R}$ is a real analytic function.

See [@CL-SIL] for a proof.

Let $f\in C^{\alpha}(\Omega)$ be a Hölder continuous potential, $\nu_f$ be the measure given by Theorem \[teo-RPF-compacto\] and $\varphi, \psi\in C^{\alpha}(\Omega)$. For each $n\in \mathbb{Z}$ we define the correlation function $$\label{correlation-function}
C_{\varphi,\psi,m_f}(n)
=
\int (\varphi\circ \sigma^n)\psi \, dm_f
-\int \varphi \,dm_f \int \psi\, dm_f.$$ We have that the above correlation function has exponential decay. More precisely we have the following proposition:

\[Decay Correlation\] For each $n\in \mathbb{N}$ let $C(n)$ denote the correlation function defined by . Then there exist constants $K>0$ and $0<\tau<1$ such that $|C(n)|\leq K \tau^n$.

The proof when $M$ is finite is given in [@Baladi]. Due to Theorem \[teo-RPF-compacto\] this proof can be easily adapted to the case where $M$ is compact metric space. We will include the details here for completeness. Before prove the above theorem, we present two auxiliary lemmas.

\[lema-proj-spectral-pif\] Let $f\in C^{\alpha}(\Omega)$, and $\partial D$ the boundary of a disc $D$ with center in $\lambda_f,$ then the spectral projection $$\pi_{f}\equiv \pi_{\mathscr{L}_{f}}=\int_{\partial D}(\lambda I-\mathscr{L}_{f})^{-1}d\lambda.$$ is given by $
\pi_{f}(\varphi) 
= 
\big(\int \varphi\, d\nu_{f}\big) \cdot h_{f}.
$

\[lema-phi1-fora\] Let be $\varphi,\psi \in C^{\alpha}(\Omega)$ then $
\mathscr{L}_{f}^n(\varphi\circ \sigma^n \cdot \psi \cdot h_{f})
=
\varphi \mathscr{L}_{f}^n( \psi h_{f}). 
$

The proof of both lemmas are straightforward computation so they will be omitted.

[**Proof of Proposition \[Decay Correlation\]**]{}. Since $m_f=h_{f} d\nu_{f}$ it follows from the definition of the correlation function that $$\begin{aligned}
\label{D. Correlation 2}
|C_{\varphi,\psi,m_f }(n)|
&=
\left| 
\int (\varphi\circ \sigma^n)\psi h_{f}\, d\nu_{f}
- 
\int\varphi  h_{f}\, d\nu_{f}
\int\psi h_{f}\, d\nu_{f}                            
\right|.\end{aligned}$$ Notice that $
(\mathscr{L}^{*}_{f})^n\nu_{f}
=
\lambda_{f}^n\nu_{f}
$ and therefore the rhs above is equal to $$\left|
\int\lambda_{f}^{-n}
\mathscr{L}_{f}^n((\varphi\circ \sigma^n)\psi h_{f})
\, d\nu_{f}
-
\int\varphi  h_{f}\, d\nu_{f}
\int\psi h_{f}\,  d\nu_{f}                            
\right|.$$ By using the Lemma \[lema-phi1-fora\] and performing simple algebraic computations we get $$\begin{aligned}
 \label{estimativa-1-dec-pol}
|C_{\varphi,\psi,m_f }(n)|
\leq 
\left(   \int|\varphi|\, d\nu_{f}    \right)
\left\|
\lambda^{-n}_{f} \mathscr{L}_{f}^n
\left(
    \psi h_{f}-  h_{f}\int\psi h_{f}\, d\nu_{f}  
\right) 
\right\|_{0}.\end{aligned}$$

By Theorem \[teo-RPF-compacto\] we know that the spectrum of $\mathscr{L}_{f}: C^{\alpha}(\Omega)\to C^{\alpha}(\Omega)$ consists in a simple eigenvalue $\lambda_{f}>0$ and a subset of a disc of radius strictly smaller than $\lambda_{f}.$ Set $
\tau=\sup\{|z|; |z|<1~\textnormal{and}~z\cdot\lambda_{f} \in 
\mathrm{Spec}(\mathscr{L}_{f})\}.
$ The existence of the spectral gap guarantees that $\tau<1$. Let $\pi_{f}$ the spectral projection associated to eigenvalue $\lambda_{f}$, then, the spectral radius of the operator $\mathscr{L}_{f}(I-\pi_{f})$ is exactly $\tau\cdot \lambda_f$. Since the commutator $[\mathscr{L}_{f},\pi_{f}]=0$, we get $\forall n \in\mathbb{N}$ that $
[\mathscr{L}_{f}(I-\pi_{f})]^n
=
\mathscr{L}_{f}^{n}(I-\pi_{f}).
$ From the spectral radius formula it follows that for each choice of $\widetilde{\tau}>\tau$ there is $n_0\equiv n_0(\widetilde{\tau})\in\mathbb{N}$ so that for all $n\geq n_0$ we have $
\|\mathscr{L}_{f}^{n}(\varphi-\pi_{f}\varphi)\|_0
\leq 
\lambda_{f}^n\widetilde{\tau}^n
\|\varphi\|_0,~\forall\varphi\in  C^{\alpha}(\Omega).
$ Therefore there is a constant $C( \widetilde{\tau})>0$ such that for every $n\geq 1$ $$\|\mathscr{L}_{f}^n(\varphi-\pi_{f}\varphi)\|_0
\leq 
C( \widetilde{\tau}) \ \lambda_{f}^n\ \widetilde{\tau}^n\ \|\varphi\|_0
\qquad\forall\varphi\in  C^{\alpha}(\Omega).$$ By using the Lemma \[lema-proj-spectral-pif\] and the above upper bound in the inequality we obtain $$\begin{aligned}
|C_{\varphi,\psi,m_f}(n)|
&\leq
\left( \int |\varphi|\, d\nu_{f} \right)
C\ \widetilde{\tau}^n\ \|\psi h_{f}\|_0
\\
&\leq
C(\widetilde{\tau}) 
\|h_{f}\|_{0} 
\left( \int |\varphi|d\nu_{f} \right)
\|\psi\|_0 \ \widetilde{\tau}^n.
\tag*{\qed} \end{aligned}$$

Main Results
============

Proposition \[Principio variacional\] ensures for any Hölder potential $f$ that the limit $P(f)
=
\lim_{n\to\infty}
n^{-1}
\log[ \mathscr{L}_{f}^n({\bf 1})(x)]
$ always exist and is independent of $x\in \Omega$. In what follows, we will extend this result for all continous potentials $f\in C(\Omega)$. This is indeed a surprising result and it does not have a counter part on one-dimensional two-sided lattices, due to the existence of metastable states discovered by Sewell in [@Sewell]. Before present the proof of this fact, we want to explain what is the mechanism behind the absence of metastable states for one-dimensional systems on the lattice $\mathbb{N}$. The absence of metastable states for continuous potentials and finite state space $M$, on one-dimensional one-sided lattices, as far as we known was first proved by Ricardo Mañé in [@ma], but apparently he did not realize it. A generalization of this result for $M$ being a compact metric space appears in [@CL16] and again no mention to metastable states is made in this paper. The proof of this result presented in [@CL16] is completely different from ours and we believe that one presented here is more suitable to be adapted to other context. An alternative explanation of this fact can be found in [@CER17] Remark 2.2, and following the first author of [@CER17] the first person to realize this fact was Aernout van Enter.

Let $(\widetilde{\mathscr{B}},\|\cdot\|)$ and $(\mathscr{B},|\!|\!|\cdot|\!|\!| )$ the classical Banach spaces of interactions defined as in [@Israel]. In case of free boundary conditions, we have for any $\Phi\in\widetilde{\mathscr{B}}$ and a finite volume $\Lambda_n\subset\mathbb{Z}$ that $
|P_{\Lambda_n}(\Phi)-P_{\Lambda_n}(\Psi)| \leq |\!|\!|\Phi-\Psi|\!|\!|
$ and therefore the finite volume pressure with [**free boundary conditions**]{} is 1-Lipschitz function from $(\widetilde{\mathscr{B}},|\!|\!|\cdot|\!|\!|)$ to $\mathbb{R}$. On the lattice $\mathbb{Z}$, when boundary conditions are considered the best we can prove is $
|\tau_nP_{\Lambda_n}(\Phi)-\tau_nP_{\Lambda_n}(\Psi)|
\leq 
\|\Phi-\Psi\|,
$ for interactions $\Phi,\Psi\in \mathscr{B}$. From this we have that finite volume pressure with boundary conditions is 1-Lipschitz function from the smaller Banach space $(\mathscr{B},\|\cdot\|)$ to $\mathbb{R}$. The last inequality can not be improved for general boundary conditions and interactions in the big Banach space $(\widetilde{\mathscr{B}},|\!|\!|\cdot|\!|\!|)$, because as we will see in the proof of Theorem \[Pressao-assintotica\] it would imply the independence of the boundary conditions of the infinite volume pressure which is a contradiction with Sewell’s theorem. The mechanism that prevents existence of metastable states for continuous potential on the lattice $\mathbb{N}$ is the possibility of proving that the analogous of the finite volume pressure with boundary conditions on the lattice $\mathbb{N}$ is indeed 1-Lipschitz function. To be more precise.

\[Pressao-assintotica\] For each continuous potential $f\in C(\Omega)$ there is a real number $P(f)$ such that $$\lim_{n\to \infty}
\left\| \dfrac{1}{n}\log\mathscr{L}_f^n{\bf 1} -P(f)\right\|_{0}
=0.$$

It is sufficient to prove that $\Phi_{n}:C(\Omega) \to C(\Omega)$, given by $$\Phi_{n}(f)(x)=\frac{1}{n}\log \mathscr{L}_f^n{\bf 1}(x)$$ converges to a Lipschitz continuous function $\Phi:C(\Omega) \to C(\Omega)$ in the following sense $\|\Phi_n(f)-\Phi(f)\|_{0}\to 0$, when $n\to\infty$. Indeed, by Proposition \[Principio variacional\], for any fixed $0<\alpha< 1$ we have $\Phi(C^{\alpha}(\Omega))\subset \langle 1\rangle$, where $\langle 1 \rangle$ denotes the subspace generated by the constant functions in $C(\Omega)$. Since $C^{\alpha}(\Omega)$ is a dense subset of $(C(\Omega),\|\cdot\|_{0})$ and $\Phi$ is Lipschitz, then $\Phi(C(\Omega))=\Phi(\overline{C^{\alpha}(\Omega)})\subset \langle 1 \rangle$.

In order to deduce the convergence of $
(\Phi_{n})_{n\in \mathbb{N}},
$ it is more convenient to identify $
\Phi_{n}:C(\Omega)\to C(\Omega)
$ with the function $
\Phi_{n}:C(\Omega)\times \Omega\rightarrow\mathbb{R},
$ given by $
\Phi_{n}(f,x)=(1/n)\log \mathscr{L}_f^n{\bf 1}(x).
$ For any fixed $x\in\Omega$, follows from the Dominated Convergence Theorem that the Fréchet derivative of $
\Phi_{n}:C(\Omega)\times \Omega\rightarrow\mathbb{R},
$ evaluated at $f$ and computed in $\varphi$, is given by $$\frac{\partial}{\partial f}\Phi_n(f,x)\cdot\varphi
=
\frac{1}{n}
\frac{\displaystyle\int_{M^n}(S_{n}\varphi)({{\bf a}}x)\exp(S_{n}f)({{\bf a}}x)\, d\mu({{\bf a}})}
{\mathscr{L}_f^n{\bf 1}(x)},$$ where $(S_{n}\varphi)(x)\equiv  \sum_{j=0}^{n-1}\varphi\circ\sigma^{j}(x).$ Clearly, $
\displaystyle\left\| n^{-1}S_n\varphi\right\|_{0}
\leq
\|\varphi\|_{0},
$ and therefore we have the following estimate $$\begin{aligned}
\label{pressao 3}
\left|\frac{\partial}{\partial f}\Phi_{n}(f,x)\cdot\varphi\right|
&=
\left|\frac{\displaystyle\int_{M^n}\frac{1}{n}(S_{n}\varphi)({{\bf a}}x)\exp(S_{n} f)({{\bf a}}x)\, d\mu({{\bf a}})}
{\mathscr{L}_f^n{\bf 1}(x)}\right|
\nonumber
\\
&\leq
\frac{\displaystyle\int_{M^n}\left|\frac{1}{n}(S_{n}\varphi)({{\bf a}}x)\exp(S_{n}f)({{\bf a}}x)\, \right| d\mu({{\bf a}})}
{\left|\displaystyle\int_{M^n}\exp(S_{n}f)({{\bf a}}x)\, d\mu({{\bf a}})\right|}
\nonumber
\\
&\leq 
\|\varphi\|_0,\end{aligned}$$ for any $f\in C(\Omega)$ independently of $x\in \Omega$. Taking the supremum in over $x\in\Omega$ we obtain $$\left\|\frac{\partial}{\partial f}\Phi_{n}(f,\cdot)\cdot\varphi\right\|_0
\leq 
\|\varphi\|_0,$$ for all $n\in\mathbb{N}$. The above inequality allows to conclude that $\|\frac{\partial}{\partial f}\Phi_{n}(f)\| \leq 1$, where $\|\cdot\|$ means the operator norm.

Fix $f$ and $\tilde{f}$ in $C(\Omega)$ and define for each $n\in \mathbb{N}$ the map $\hat{\Phi}_{n}(t)=\Phi_{n}(\alpha(t),x),$ where $\alpha(t)=t f+(1-t)\tilde{f}$ with $0\leq t\leq 1$. Obviously, $\hat{\Phi}_{n}$ is a differentiable map when seen as a map from $[0,1]$ to $\mathbb{R}$ and we have that $
|\hat{\Phi}_{n}(1)-\hat{\Phi}_{n}(0)|=|\frac{d}{dt}\hat{\Phi}_{n}(\hat{t})(1-0)|
$ for some $\hat{t}\in (0,1)$. Using the above estimative of the Fréchet derivative norm we have that $$\begin{aligned}
\label{estimativa-dif-Phin}
|\Phi_{n}(f,x)-\Phi_{n}(\tilde{f},x)|
=
\left|\frac{\partial}{\partial f}\Phi_{n}(f,x)(f-\tilde{f})\right|\leq\|f-\tilde{f}\|_{0}.\end{aligned}$$ As an outcome for any fixed $f\in C(\Omega)$ the sequence $(\Phi_{n}(f))_{n\in\mathbb{N}}$ is uniformly equicontinuous. Moreover, $\sup_{n\in\mathbb{N}}\|\Phi_{n}(f)\|_{0}<\infty$. Indeed, from inequality $\eqref{estimativa-dif-Phin}$, the triangular inequality and existence of the limit $\lim_{n\to\infty} \Phi_n({\bf 1})$, it follows that $$\begin{aligned}
 |\Phi_n(f)|
 =
 |\Phi_n(f)-\Phi_n({\bf 1})|+|\Phi_n({\bf 1})|
 &\leq 
 \|f-{\bf 1} \|_{\infty} +|\Phi_n({\bf 1})|
 \\
 &\leq
 \|f-{\bf 1}\|_{\infty} +\sup_{n\in\mathbb{N}}|\Phi_n({\bf 1})|
 \\
 &\equiv 
 M(f).
 \end{aligned}$$ Now, we are able to apply the Arzelà-Ascoli’s Theorem to obtain a subsequence $
(\Phi_{n_k}(f))_{k\in\mathbb{N}},
$ which converges to a function $\Phi(f)\in C(\Omega)$.

We now show that $\Phi_n(f)\to \Phi(f)$, when $n\to\infty$. Let $\varepsilon>0$ and $g\in C^{\alpha}(\Omega)$ such that $\|f-g\|_0<\varepsilon$. Choose $n_k$ and $n$ sufficiently large so that the inequalities $\|\Phi_{n_k}(f)- \Phi(f)\|_{0}<\varepsilon$, $\|\Phi_n(g)-\Phi_{n_k}(g)\|_{0}<\varepsilon$ and $\|\Phi_{n_k}(g)-\Phi_{n_k}(f)\|_{0}<\varepsilon$ are satisfied. For these choices of $g,n$ and $n_k$, we have by the triangular inequality and inequality that $$\begin{aligned}
\|\Phi_n(f)- \Phi(f)\|_{0}
&\leq 
\|\Phi_n(f)-\Phi_{n_k}(f)\|_{0}
+
\|\Phi_{n_k}(f)- \Phi(f)\|_{0}
\\
&<
\|\Phi_n(f)-\Phi_{n_k}(f)\|_{0}
+
\varepsilon
\\
&\leq 
\|\Phi_n(f)-\Phi_{n}(g)\|_{0}
+
\|\Phi_n(g)-\Phi_{n_k}(f)\|_{0}
+
\varepsilon
\\
&< 
\|\Phi_n(g)-\Phi_{n_k}(f)\|_{0}
+
2\varepsilon
\\
&\leq 
\|\Phi_n(g)-\Phi_{n_k}(g)\|_{0}
+
\|\Phi_{n_k}(g)-\Phi_{n_k}(f)\|_{0}
+
2\varepsilon
\\
&<
4\varepsilon,\end{aligned}$$ thus proving the desired convergence.

To finish the proof it is enough to observe that the inequality \[estimativa-dif-Phin\] implies that $\Phi$ is a Lipschitz continuous function.

\[Pressure derivative\] For each fixed $0<\alpha<1$ and $f\in C^{\alpha}(\Omega)$ the Fréchet derivative of the pressure functional $P:C^{\alpha}(\Omega)\rightarrow\mathbb{R}$ is given by $$\label{pressure-derivative}
P'(f)\varphi=\int\varphi h_{f}\, d\nu_{f}$$

For each fixed $f \in  C^{\alpha}(\Omega),$ $0<\alpha< 1,$ there exists the limit $$\lim_{n\to \infty}
\left\|
\frac{1}{n}
\frac{\mathscr{L}_{f}^{n}(S_{n}\varphi)}
{\mathscr{L}^{n}_{f}{\bf 1}}
-
\int\varphi h_{f}\, d\nu_{f}
\right\|_0
=
0$$ for every $\varphi\in C(\Omega)$, and the convergence is uniform.

A straightforward calculation shows that $$\begin{aligned}
\label{desigualdade}
\left\|
\frac{1}{n}\frac{\mathscr{L}_{f}^{n}(S_{n}\varphi)}{\mathscr{L}^{n}_{f}{\bf 1}}
-
\int\varphi h_{f}\, d\nu_{f}
\right\|_{0}
&=
\left\|
\frac{\lambda_f^{n}}{\mathscr{L}^{n}_{f}{\bf 1}}\frac{1}{n}
\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi)
-
\int\varphi h_{f}\, d\nu_{f}
\right\|_{0}
\\[0.4cm] 
&
\hspace*{-3,175cm}
\leq
\sup_{n\in\mathbb{N}}
\left\|\frac{\lambda_f^{n}}{\mathscr{L}^{n}_{f}{\bf 1}}\right\|_{0}
\left\|\frac{1}{n}\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi)-
(\lambda_f^{-n}\mathscr{L}^{n}_{f}{\bf 1})\int\varphi h_{f}\, d\nu_{f}\right\|_{0}.
\nonumber\end{aligned}$$ Therefore to get the desired result it is sufficient to show that:

-   $
    \displaystyle\sup_{n}\left\|\frac{\lambda_f^{n}}{\mathscr{L}^{n}_{f}{\bf 1}}\right\|_{0} 
    $ is finite;

-   $
    \displaystyle \quad (\lambda_f^{-n}\mathscr{L}^{n}_{f}{\bf 1})\int\varphi h_{f}\, d\nu_{f}
    $ converges to $
    \displaystyle h_f\int\varphi h_{f}\, d\nu_{f};
    $

-   $
    \displaystyle \quad\frac{1}{n}\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi) $ converges to $
    \displaystyle h_f\int\varphi h_{f}\, d\nu_{f}.
    $

The first two items are immediate consequences of the Ruelle-Perron-Frobenius Theorem. Indeed, the convergence $
\lambda_f^{-n}\mathscr{L}_f^n {\bf 1}\stackrel{\|\cdot\|_0}{\longrightarrow}h_f,
$ immediately give that $$\begin{aligned}
(\lambda_f^{-n}\mathscr{L}^{n}_{f}{\bf 1})
\int\varphi h_{f}\, d\nu_{f} 
\stackrel{\|\cdot\|_0}{\longrightarrow}  
h_f\int\varphi h_{f}\, d\nu_{f}.\end{aligned}$$ Since $h_f$ is a continuous strictly positive function, it follows from compactness of $\Omega$ that $h_f$ is bounded away from zero, and consequently $
 \lambda_f^{n}/\mathscr{L}_f^n{\bf 1} \stackrel{\|\cdot\|_0}{\longrightarrow} 1/h_f.
 $ Once $h_f$ is strictly positive $1/h_f$ is also positive and bounded away from zero, which gives that $$\begin{aligned}
\sup_{n}\left\|\frac{\lambda_f^{n}}{\mathscr{L}^{n}_{f}{\bf 1}}\right\|_{0}<\infty.\end{aligned}$$

The third expression in (c) is harder to analyze than the previous two, so we will split the analysis in three claims.

*Claim 1.* For all $\varphi\in C(\Omega)$ and $n\in\mathbb{N}$ we have that, $$\label{Claim 1}
\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi)
=
\sum_{j=0}^{n-1}\lambda_f^{-(n-j)}\mathscr{L}_{f}^{n-j}
(\varphi\lambda_f^{-j}\mathscr{L}^{j}_{f}{\bf 1}).$$

We first observe that, $
\mathscr{L}_{f}^{n}(\varphi\circ\sigma^{n})
=
\varphi\mathscr{L}^{n}_{f}{\bf 1},
$ which is an easy consequence of the definition of the Ruelle operator. From that and the linearity of the Ruelle operator it follows, $$\begin{aligned}
\mathscr{L}_{f}^{n}(S_{n}\varphi)
&=
\mathscr{L}_{f}^{n}(\varphi)+\mathscr{L}_{f}^{n-1}(\mathscr{L}_{f}(\varphi\circ\sigma))+
\ldots+\mathscr{L}_{f}(\mathscr{L}_{f}^{n-1}\varphi\circ\sigma^{n-1})
\\
&
\hspace*{-1cm}
=
\mathscr{L}_{f}^{n}(\varphi)+\mathscr{L}_{f}^{n-1}(\varphi\mathscr{L}_{f}{\bf 1})+
\ldots+\mathscr{L}_{f}(\varphi\mathscr{L}^{n-1}_{f}{\bf 1})
=
\sum_{j=0}^{n-1}\mathscr{L}_{f}^{n-j}(\varphi\mathscr{L}^{j}_{f}{\bf 1})\end{aligned}$$ finishing the proof. From the linearity of the Ruelle operator we easily get $$\begin{aligned}
\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi)
=
\sum_{j=0}^{n-1}\lambda_f^{-(n-j)}\mathscr{L}_{f}^{n-j}
(\varphi\lambda_f^{-j}\mathscr{L}^{j}_{f}{\bf 1}).\end{aligned}$$

*Claim 2.* For each $\varphi\in C(\Omega)$ we have $$\label{eq:5}
\lim_{n\to \infty}\left\|
\frac{1}{n}\left(
\lambda_f^{-n}\mathscr{L}_{f}^{n}(S_{n}\varphi)-\sum_{j=0}^{n-1}\lambda_f^{-(n-j)}\mathscr{L}_{f}^{n-j}\varphi h_{f}
\right)
\right\|_{0}
=
0.$$ To verify we use to obtain the following estimate, $$\begin{aligned}
\left\|
    \frac{1}{n}
    \left\{
        \lambda_f^{-n}\mathscr{L}_f^{n}(S_{n}\varphi)
        -
        \sum_{j=0}^{n-1}\lambda_f^{-(n-j)}\mathscr{L}_f^{(n-j)}\varphi h_f
    \right\}
\right\|_{0}
\\[0.4cm]
&\hspace*{-4cm}=
\left\|\frac{1}{n}\sum_{j=0}^{n-1}\lambda_f^{-(n-j)}\mathscr{L}_f^{n-j}\left(\varphi\lambda_f^{-j}\mathscr{L}^{j}_f{\bf 1}-\varphi h_f\right)\right\|_{0}
\\[0.4cm]
&\hspace*{-4cm}\leq
\frac{const.}{n}\, 
\sum_{j=0}^{n-1}\left\|\varphi\lambda_f^{-j}\mathscr{L}^{j}_f{\bf 1}-\varphi h_f\right\|_{0}.\end{aligned}$$ The last term in the above inequality converges to zero, when $n\to \infty$, because it is a Cesàro summation associated to the sequence $\varphi\lambda_f^{-j}\mathscr{L}^{j}_f{\bf 1}-\varphi h_f$, which converges to zero in the uniform norm by the Ruelle-Perron-Frobenius Theorem, so the claim is proved.

*Claim 3.* For each $\varphi\in C(\Omega)$ there exists the following limit $$\label{eq:7}
\lim_{n\to \infty}\left\|\frac{1}{n}\sum_{j=0}^{n-1}\lambda_{f}^{-(n-j)}\mathscr{L}_{f}^{n-j}\varphi h_{f}-
h_{f}\int\varphi h_{f}d\nu_{f}\right\|_{0}=0.$$ Define $A_{n,j}:=\lambda_f^{-(n-j)}\mathscr{L}_{f}^{n-j}\varphi h_{f}$ and $B:=h_{f}\int\varphi h_{f}\, d\nu_{f}.$ In one hand, by the Ruelle-Perron-Frobenius Theorem, we must have for any fixed $j\in\mathbb{N}$ that $\lim_{n\to \infty}\left\|A_{n,j}-B\right\|_{0}=0.$ On the other hand, we have by the triangular inequality and the convergence in the Cesàro sense that $$\begin{aligned}
\left\|\dfrac{1}{n}\sum_{j=0}^{n-1}A_{n,j}-B\right\|_{0}
\leq
\dfrac{1}{n}\sum_{j=0}^{n-1}\left\|A_{n,j}-B\right\|_{0}\longrightarrow 0,\end{aligned}$$ when $n\rightarrow\infty,$ finishing the proof of Claim 3. Therefore the proof of the Lemma is established.

[**Proof of Theorem \[Pressure derivative\]**]{}

Fix $x\in \Omega $ and define function $\Phi_{n}:C^{\alpha}(\Omega)\rightarrow\mathbb{R}$ as $$\Phi_{n}(f)\equiv \frac{1}{n}\log(\mathscr{L}_{f}^{n}{\bf 1})(x).$$ As we have seen in the proof of Theorem \[Pressao-assintotica\] the Fréchet derivative of $\Phi_n$ at $f$ evaluated in $\varphi\in C^{\alpha}(\Omega)$ is given by $$\Phi_{n}'(f)\varphi
=
\frac{1}{n}
\frac{\mathscr{L}_{f}^{n}(S_{n}\varphi)}{\mathscr{L}^{n}_{f}{\bf 1}},$$ Since we have the analyticity of the pressure functional in $C^{\alpha}(\Omega)$ (Theorem \[Pressure Analiticity\]) it follows from the previous Lemma that $$P'(f)\varphi
=
\lim_{n\to \infty} \Phi_{n}'(f)\varphi
=
\int\varphi h_{f}\, d\nu_{f}.$$

A Heisenberg type Model 
========================

The aim of this section is to introduce a Heisenberg type model on the half-space $\mathbb{N}\times \mathbb{Z}$, prove absence of phase transition and exponential decay of correlations for this model.

The construction of this model is split in two steps. Firts step. We consider $(S^2)^\mathbb{Z}$ as the configuration space. At inverse temperature $\beta\in (0, \infty),$ the configurations are randomly chosen according to the following probabilities measures $\mu_{n, \beta}$ $$\label{Heisenberg-measure}
d\mu_{n, \beta}(\sigma):=\dfrac{1}{Z_{n, \beta}}
\exp (
\beta \sum_{i,j\in \Lambda_n} \sigma_i\cdot \sigma_j
)\, d\sigma,$$ where $\Lambda_n$ denotes the symmetric interval of integers $[-n,n]$, $\sigma_i\cdot \sigma_j$ denotes the inner product in $\mathbb{R}^3$ of the first neighbors $\sigma_i$ and $\sigma_j$, $$Z_{n, \beta}=\int_{(S^2)^{\Lambda_n}}\exp
(
\beta \sum_{i,j} {\sigma_i\cdot \sigma_j }
)\, d\sigma$$ and $d\sigma$ is the uniform probability measure on $(S^{2})^{\Lambda_n}$. Let measure $\hat{\nu}$ be the unique accumulation point of the sequence of probability measures given by , see [@Geogii88] for details. The measure $\hat{\nu}$ will be used as the a priori measure in the second step.

Second step. Now we introduce a Heisenberg type model. We begin with the compact metric space $(S^2)^{\mathbb{Z}}$, where $S^2$ is the 2-dimensional unit sphere in $\mathbb{R}^3$, as our alphabet. Now the configuration space is the Cartesian product $\Omega =((S^2)^{\mathbb{Z}})^{\mathbb{N}},$ that is, a configuration is a point $\sigma=(\sigma(1),\sigma(2), \cdots)\in \Omega$, where each ${\sigma}(i)$ is of the form $\sigma(i)=(\ldots,\sigma_{(i,-2)},\sigma_{(i,-1)},\sigma_{(i,0)}, \sigma_{(i,1)}, \ldots)$, and each $\sigma_{(i,j)}\in S^2$. We denote by $\|\cdot\|$ the $\mathbb{R}^3$ Euclidean norm, and $v\cdot w$ the inner product of two elements of $\mathbb{R}^3$.

Fix a summable ferromagnetic translation invariant interaction $J$ on $\mathbb{Z}$, that is, a function $J:\mathbb{Z}\to (0,\infty)$ and assume that $J(n)= e^{-|n|\alpha}$, for some $\alpha>0$. Of course, we have $
\sum_{n\in \mathbb{Z}} J(n)<\infty.
$ Now we consider the potential $f:\Omega \to \mathbb{R}$ given by $$\label{Heisenberg Potential}
f(\sigma)= \sum_{ n\in \mathbb{Z}} J(n)\ \sigma_{(1,n)}\cdot\sigma_{(2,n)}.$$ Note that this potential has only first nearest neighbors interactions.

![The configuration space $((S^2)^{\mathbb{Z}})^{\mathbb{N}}$[]{data-label="fig:FigPaper"}](FigPaper2){width="0.6\linewidth"}

The potential $f$ given by is actually an $\alpha$-Hölder continuous function. Indeed, $$\begin{aligned}
\label{Holder-1}
|f(\sigma)-f(\omega)|
&\leq 
\left |\sum_{n\in \mathbb{Z}}J(n)\sigma_{(1,n)}\cdot (\sigma_{(2,n)}-\omega_{(2,n)})\right|\nonumber
\\
&\hspace*{3.15cm} +
\left|\sum_{n\in \mathbb{Z}}J(n)\omega_{(2,n)}\cdot (\sigma_{(1,n)}-\omega_{(1,n)})\right|
\nonumber
\\[0.5cm]
&\leq
\sum_{n\in \mathbb{Z}}J(n)\|\sigma_{(2,n)}-\omega_{(2,n)}\|+ 
\sum_{n\in \mathbb{Z}}J(n)\|\sigma_{(1,n)}-\omega_{(1,n)}\| \end{aligned}$$

From the very definition of the distance we have

$$\begin{aligned}
\label{Holder-2}
d(\sigma,\omega)
\geq 
\dfrac{1}{2^n}\sum_{j\in \mathbb{Z}}\dfrac{1}{2^{|j|}}\|\sigma_{(n,j)}-\omega_{(n,j)}\|
\geq 
\dfrac{1}{2^{n+|j|}}\|\sigma_{(n,j)}-\omega_{(n,j)}\|.\end{aligned}$$

By using and we get that $$\dfrac{|f(\sigma)-f(\omega)|}{d(\sigma,\omega)^{\alpha}}
\leq  
K_1\sum_{n\in \mathbb{N}}J(n)2^{\alpha n}
+
K_2\sum_{n\in \mathbb{N}}J(n)2^{\alpha n}.$$ Since for all $n\in\mathbb{N}$ we have $
J(n)2^{\alpha n}
\leq 
\exp({-n(\alpha(1-\log 2))}),
$ and the constant $\alpha (1-\log 2)$ is positive, follows that the series $\sum_{n\in \mathbb{N}}J(n)2^{\alpha n}$ is convergent. Therefore, $f$ is an $\alpha$-Hölder continuous function.

From Theorem \[teo-RPF-compacto\], we have that there is a unique probability measure $\nu_{f}$ so that $\mathscr{L}_{f}^{*}\nu_f=\lambda_{f}\nu_f$. This probability measure, following [@CL16], is a unique DLR-Gibbs measure associated to a quasilocal specification associated to $f$, see [@CL14] for the construction of this specification.

By observing that the horizontal interactions, in our model, goes fast to zero in the $y$-direction is naturally to expect that the model is essentially a one-dimensional model. This feature allow us to obtain the following result

Let $\beta f$ be a potential, where $f$ is given by and the inverse temperature $\beta \in (0,\infty)$. Consider the a priori measure $\hat{\nu}$ on $(S^2)^{\mathbb{Z}}$, constructed from . Then for any fixed $\beta>0$ and $m\in\mathbb{Z}$ there are positive constants $K(\beta)$ and $c(\beta)$ such that for all $n\in\mathbb{N}$ we have $$\int_{\Omega} (\sigma_{(1,m)}\cdot\sigma_{(n+1,m)})\ d\nu_{\beta f}
\leq 
K(\beta)e^{-c(\beta)n}.$$ Furthermore, the pressure functional is differentiable at $\beta f$ and its derivative is given by expression .

Fix $m\in\mathbb{Z}$ and let $\sigma_{(n,m)}\equiv(\sigma_{(n,m)}^{x},\sigma_{(n,m)}^{y}, \sigma_{(n,m)}^{z})$. Consider the following continuous potentials $\varphi^u, \xi^u$ given by $$\varphi^u(\sigma) = \sigma_{(1,m)}^{u}
\qquad\text{and}\qquad
\xi^u(\sigma) = \frac{\sigma_{(1,m)}^{u}}{h_{\beta f}(\sigma)}, \quad u=x,y,z.$$ Note that $$\int_{\Omega} \xi \, dm_{\beta f}
=
\int_{\Omega} \sigma_{(1,m)}^{u} \, d\nu_{\beta f}(\sigma) 
=
0,\quad u=x,y,z,$$ where the last equality comes from the $O(3)$-invariance of the eingemeasure.

Since $h_{\beta f}(\sigma)=h_{\beta f}(-\sigma)$, see [@CL17], it follows again from the $O(3)$-invariance of $\nu_{f}$ that $$\int_{\Omega} \sigma_{(1,m)}^{u} \, dm_{\beta f} 
= 
\int_{\Omega} \sigma_{(1,m)}^{u} h_f(\sigma) \, d\nu_{\beta f} 
=
0, \quad u=x,y,z.$$ Therefore $$\begin{aligned}
\label{Decay-Two-Point}
\int_{\Omega} (\sigma_{(1,m)}^{u}\sigma_{(n+1,m)}^u)\ d\nu_{\beta f}
&=
\int (\varphi^u\circ \sigma^n)\xi^u \, dm_{\beta f}
\nonumber
\\
&=
C_{\varphi^{u},\xi^{u},m_{\beta f}}(n)
=
O(e^{-c(\beta)n}),  \end{aligned}$$ $u=x,y,z.$ By summing with $u=x,y,z$ we get the claimed exponential decay.

The last statement follows from the fact that $(S^2)^{\mathbb{Z}}$ is a compact metric space, $\hat{\nu}$ is a full support a priori measure in $(S^2)^{\mathbb{Z}}$ and $\beta f$ is an $\alpha$-Hölder. Therefore the Proposition \[Decay Correlation\] applies and corollary follows.

Acknowledgments {#acknowledgments .unnumbered}
===============

The author would like to thank Leandro Cioletti, Artur Lopes and Andréia Avelar for fruitfull discussions and comments.

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