Markov approximations of Gibbs measures for long-range interactions on 1D lattices

We consider a one-dimensional infinite lattice. At each site of the lattice is placed a particle that interacts with the other particles via pair interactions. In order to model such pair interactions between particles let us consider a finite set A, called alphabet, representing the possible 'states' of the particle. As usual, the interaction of every pair of particles will depend on its specific states and the distance between them. Let X = A^N₀ be the space of all infinite sequences made up of symbols taken from A, in other words, the set of all possible configurations of our system. A word of length k (k∈N₀) is the finite sequence ${a}_{1}^{k}:= {a}_{1}{a}_{2}\cdots {a}_{k}$ where a_i∈A for all i = 1,...,k. We endow A with the discrete topology. The space X is supplied with the product topology, and thus it is compact. We consider the Borel sigma algebra generated by the cylinder sets defined by $[{a}_{0}^{k}]:= \{ x\in X~:~{x}_{0}^{k}={a}_{0}^{k}\}$.

Next, the dynamics in X is generated by the shift map T, that is, (Tx)_i = x_i+1 for any x∈X. Let _T(X) be the space of all T-invariant probability measures on X. Consider the following distance on _T(X):

$$\begin{eqnarray*} &&D(\mu ,\nu ):= \sum _{k=0}^{\infty }{2}^{-k+1}\left (\mathop{\sum }\limits _{a\in {A}^{k}}\vert \mu [a]-\nu [a]\vert \right ). \end{eqnarray*}$$

It is well known that this distance is a metric for the weak-∗ topology and on the same topology _T(X) is a compact and convex set.

Notational remark: From now on, we denote by a = c^±1 the inequalities c⁻¹ ≤ a ≤ c. Analogously a = exp(±b) stands for exp(−b) ≤ a ≤ exp(b).

The following definition will be important for stating our results.

Definition 2.1 (Projective convergence⁴ ) Let r,n∈N₀. The sequence of measures {ν_r} is said to converge in the projective sense to ν if there exists a sequence {ε_r}, such that ε_r → 0 as r → ∞ for any cylinder $[{a}_{0}^{n-1}]$, for which the following inequalities hold:

$$\begin{eqnarray*} &&\frac{{\nu }_{r}[{a}_{0}^{n-1}]}{\nu [{a}_{0}^{n-1}]}={\mathrm{e}}^{\pm n{\varepsilon }_{r}}. \end{eqnarray*}$$

We are interested in quantifying the 'energy' of a given configuration x. The local energy (centred at 0) of the configuration x is a function u:X → R defined as follows:

$$\begin{eqnarray} &&u(x):= \sum _{k=0}^{\infty }{\psi }_{k}({x}_{0},{x}_{k}), \end{eqnarray} \tag{ 1 }$$

where ψ_k:A × A → R is a distance-dependent pair-wise interaction. A classical example is an Ising-type model studied by Dyson, for which A = {1, − 1} and the pair-wise interaction is ψ_k(x₀,x_k): =− βJ(x₀x_k)/(k^α), where β is the inverse temperature and J is a 'coupling constant'. It is known that, for α > 2, the Bowen–Walters condition holds and thus there is a unique equilibrium state [11].

We will assume the following condition.

Condition 1 Let p:R → R be a function such that p(r)/r → ∞ as r → ∞, we consider pair-wise interactions ψ_k satisfying

$$\begin{eqnarray*} &&\sum _{r=0}^{\infty }p(r)\Vert {\psi }_{r}\Vert \lt \infty . \end{eqnarray*}$$

Remark 2.1 Observe that this condition is slightly stronger than the Bowen–Walters condition and the Ruelle condition [14]. We emphasize the advantage of considering this condition instead.

The states of the systems are described by probability measures. The probability measure μ∈_T(X) is said to be an equilibrium state for u if

$$\begin{eqnarray*} &&h(\mu )+\int u \hspace{0.167em} \mathrm{d}\mu =\mathop{\sup }\limits _{\nu \in {\mathcal{M}}_{T}(X)}\left \{ h(\nu )+\int u \hspace{0.167em} \mathrm{d}\nu \right \} , \end{eqnarray*}$$

where h(ν) is the measure-theoretic entropy of ν defined as

$$\begin{eqnarray*} &&h(\nu )=-\mathop{\lim }\limits _{n\rightarrow \infty }\frac{1}{n+1}\mathop{\sum }\limits _{a\in {A}^{n+1}}\nu [a]\log (\nu [a]). \end{eqnarray*}$$

This definition is the well-known variational principle [3]. For lattice systems it is known that the measures fulfilling the variational principle are also Gibbs measures which are defined as follows [13].

Definition 2.2 Let (X,T) be the full shift and u:X → R be a continuous interaction. The measure μ associated with u is a Gibbs measure if there exist constants C > 1 and P(u) > 0 such that for all x∈X

$$\begin{eqnarray*} &&\frac{\mu [{x}_{0}^{n-1}]}{\exp ({U}_{n}(y)+n P(u))}={C}^{\pm }, \end{eqnarray*}$$

for all $y\in [{x}_{0}^{n-1}]$, and here ${U}_{n}(y):= {\mathop{\sum }\nolimits }_{k=0}^{n-1}u({T}^{k}y)$. The constant P(u) is the topological pressure, which in the thermodynamical formalism is defined as

$$\begin{eqnarray*} &&P(u)=h(\mu )+\int u \hspace{0.167em} \mathrm{d}\mu , \end{eqnarray*}$$

where μ is the measure that fulfils the variational principle.

At this point we should stress that the definitions given above are the same as those used in statistical mechanics up to sign conventions and constants. More precisely, a word a = a₀a₁...a_n stands for a configuration of n + 1 spins. The energy associated with such configuration times the inverse temperature β is then given by the function −U_n, or in other words −β_n = U_n. Along the same line, the thermodynamic entropy equals nh(μ) and minus the Helmholtz free energy times β is the topological pressure P(u), i.e., −β = nP(u). Thus, the Gibbs measure defined above says that the probability of observing a given spin configuration a is given by μ[a] or, equivalently

$$\begin{eqnarray*} &&\mu [a]\asymp \exp \left (-\beta {\mathcal{E}}_{n}-\beta \mathcal{A}\right ), \end{eqnarray*}$$

which is the well-known form in statistical mechanics of the Gibbs measure. Furthermore, the average of u with respect to μ times n is actually the thermodynamic energy times β,

$$\begin{eqnarray*} &&-n\int u \hspace{0.167em} \mathrm{d}\mu =\beta \mathcal{E}. \end{eqnarray*}$$

In this way, the variational principle states that the equilibrium state is the maximum over all invariant probability measures of minus the reduced Helmholtz free energy times β, i.e., 

$$\begin{eqnarray*} &&\mathop{\sup }\limits _{\nu \in {\mathcal{M}}_{T}(X)}\left \{ h(\nu )+\int u \hspace{0.167em} \mathrm{d}\nu \right \} =\mathop{\sup }\limits _{\nu \in {\mathcal{M}}_{T}(X)}\{ \mathcal{S}/n-\beta \mathcal{E}/n\} , \end{eqnarray*}$$

which clearly is equivalent to saying that the measure that minimizes the Helmholtz free energy  =− β⁻¹ +  is the equilibrium state.

In this work we use the same technique as the one implemented in [5] to characterize the Gibbs measures associated with infinite-range interactions. The strategy goes as follows. We will use the fact that every Gibbs measure can be obtained as the limit of a sequence of periodic measures, defined as follows.

Definition 2.3 Let p∈N₀ be a positive integer and denote by Per_p(X) the set of all periodic sequences x∈X of period p. For every n∈N₀ and p > n, the periodic measure of period p is defined by

$$\begin{eqnarray} &&{\mathcal{P}}^{(p)}[{a}_{0}^{n-1}]:= \frac{\mathop{\sum }_{y\in {\mathrm{Per}}_{p}(X)\cap [{a}_{0}^{n-1}]}{\mathrm{e}}^{{U}_{n}(y)}}{\mathop{\sum }_{y\in {\mathrm{Per}}_{p}(X)}{\mathrm{e}}^{{U}_{n}(y)}}. \end{eqnarray} \tag{ 2 }$$

In order to explicitly build up the sequence of measures having as a limit the Gibbs measure for the infinite-range interaction u:X → R, an intermediate step is necessary. We need to construct a Gibbs measure for interactions of range r. The latter will be taken as 'approximations' to the infinite-range interaction we are interested in. The r-range approximation u_r:X → R to the interaction u is defined as

$$\begin{eqnarray} &&{u}_{r}(x):= \sum _{k=0}^{r-1}{\psi }_{k}({x}_{0},{x}_{k}). \end{eqnarray} \tag{ 3 }$$

In this case, the interaction u_r admits a unique equilibrium measure on the full shift X, which is also a (r-step) Markov measure. This measure can be characterized by means of the concept of transfer matrix. Given the interaction u_r, we will denote by ${\tilde {u}}_{r}$ a function from A^r to R as ${\tilde {u}}_{r}({x}_{0},{x}_{1},\ldots ,{x}_{r-1})={u}_{r}(x)$ for every x = x₀x₁x₂⋯∈X. Then we define the transition matrix M_r:A^r × A^r → R⁺ associated with u_r, by

$$\begin{eqnarray} {M}_{r}(a,b):= \left \{ \begin{array}{@{}ll@{}} \displaystyle \exp \left ({u}_{r}(a{b}_{r-1})\right )\tqs &\displaystyle \text{if}~{a}_{1}^{r-1}={b}_{0}^{r-2},\\ \displaystyle 0\tqs &\displaystyle \text{otherwise}. \end{array}\right. \end{eqnarray} \tag{ 4 }$$

In the last expression ab_r−1 means the concatenation of the word a and the last letter of the word b. We will denote by L_r:A → R and R_r:A → R the corresponding left and right eigenvectors of M_r. Then the measure defined by

$$\begin{eqnarray*} &&{\mu }_{r}[{a}_{0}^{n-1}]:= {L}_{r}({a}_{0}^{r-1})\frac{\prod _{j=0}^{n-r}{M}_{r}({a}_{j}^{j+r-1},{a}_{j+1}^{j+r})}{{\rho }_{r}^{n-r+1}}{R}_{r}({a}_{n-r+1}^{n-1}) \end{eqnarray*}$$

for each n∈N₀ such that n ≥ r, is a shift-invariant probability measure. Moreover, μ_r is the unique Gibbs measure associated with the interaction u_r [6].

It is well known that the mixing property is equivalent to decay of the correlation of square integrable observables (see for instance [4, Chapter 4.3.]). So, the latter result gives improved bounds for the classical problem of a lattice of spins with polynomially decaying pair interactions, i.e. ψ_k =− βJx₀x_kk^−α with α > 2—in which, according to the operator theory [1, Chapter 1.4.], the two-site correlation function will decay polynomially with the distance between spins. In contrast, the projective convergence gives estimates for the correlation function as a stretched exponential with the distance between spins. The latter is in agreement with recent numerical simulations in which the observed decay of correlation is exponential (see for example the estimated correlation functions reported in [7]).

Our main contribution to the study of the Gibbs measures on one-dimensional lattices with long-range pair interactions (with a finite state space) is the improvement of previous results within the framework of thermodynamical formalism. Particularly, we have proved projective convergence of a sequence of finite-range Markov approximations to the unique Gibbs measure (Theorem 3.1) corresponding to a given interaction. We have also proved that the latter is mixing with a stretched exponential rate (Theorem 3.2). Finally, we have proved that the entropies of the finite-range approximations also converges, a result which is a direct consequence of the projective convergence (Theorem 3.3). For the particular case of polynomially decaying pair interactions, i.e. ψ_k =− βJx₀x_kk^−α (with β as the inverse temperature and J as a 'coupling' constant), it is known that for α > 2 there is a unique ergodic measure for all the values of β [10]. Since this potential complies with condition 1 for α > 2, our analysis concludes that for all β ≥ 0 the Markov approximations converge in the projective sense. Moreover, the above theorems let us obtain explicit formulae for the mixing rate and for the convergence rate of the entropy. On the other hand, for the case α ≤ 2 it is known that for some β_c, for all the values of β < β_c, the Gibbs measure μ_β (associated with the interaction βu) is the convex combination of two ergodic measures. In such a case the weak-∗ limit does exist and has two ergodic components. However, the Markov approximations cannot converge projectively by a theorem about the projective limit of ergodic measures due to [16]. Indeed, according to [16] the loss of projective convergence may be a signature of non-uniqueness of ergodic measures. They prove that if a sequence of ergodic measures converge in the projective sense then the limit is ergodic. In our work we have proved that, for the case of one-dimensional lattices with, to some extent, arbitrary pair interactions, the Markov approximations converges projectively to the unique Gibbs measure when the interaction fulfils condition 1. If condition 1 does not hold we have the possibility that the Gibbs measure might be decomposed into two or more ergodic measures. As mentioned above, in this case such a limit measure cannot be reached by a sequence of Markov approximations because they are ergodic (pure states) and the 'limit measure' is not. It would be interesting to see if for the case β > β_c and 1 < α ≤ 2 (for which the Gibbs measure is still in a pure state [12]) we can prove projective convergence of Markov approximations, a fact that could give us a criterion to detect phase transitions.

Consider two consecutive Markov approximations. We prove that they accumulate towards a limit measure as the range of the approximation diverges. The strategy is to use the periodic approximation of the Markov measure as follows. Using Lemma 4.1 for (r + 1)-range interactions, one obtains that

$$\begin{eqnarray*} &&{\mu }_{r+1}[{a}_{0}^{n-1}]={\mathcal{P}}_{r+1}^{(p)}[{a}_{0}^{n-1}]\cdot \exp \left (\pm (r+1){C}_{r+1}{\mathrm{e}}^{2{C}_{r+1}}\cdot {\eta }^{(p-q)/r+1-2}\right ), \end{eqnarray*}$$

where $\eta := 1-{\mathrm{e}}^{-2{\mathop{\sum }\nolimits }_{k=1}^{\infty }k\Vert {\psi }_{k}\Vert }$ and q:=max{n,r + 1}. For the sake of clarity, let us drop the indices on ${a}_{0}^{n-1}$. We compare two consecutive Markov measures,

$$\begin{eqnarray*} \frac{{\mu }_{r}[a]}{{\mu }_{r+1}[a]}&=&\frac{{\mathcal{P}}_{r}^{(p)}[a]}{{\mathcal{P}}_{r+1}^{(p)}[a]}\times \frac{\exp \left (\pm 1 2 r{C}_{r}{\mathrm{e}}^{2{C}_{r}}\cdot {\eta }^{(p-n)/r-2}\right )}{\exp \left (\pm 1 2(r+1){C}_{r+1}{\mathrm{e}}^{2{C}_{r+1}}\cdot {\eta }^{(p-q)/(r+1)-2}\right )}\\ &=&~\frac{{\mathcal{P}}_{r}^{(p)}[a]}{{\mathcal{P}}_{r+1}^{(p)}[a]}\times \exp \left (\pm 2 4(r+1){C}_{r+1}{\mathrm{e}}^{2{C}_{r+1}}\cdot {\eta }^{(p-q)/(r+1)-2}\right ). \end{eqnarray*}$$

In order to have a bound depending only on the interaction we compare two consecutive periodic measures. Since $\vert {U}_{p,r}(y)-{U}_{p,r+1}(y)\vert =\vert {\mathop{\sum }\nolimits }_{k=0}^{p}{\psi }_{r+1}({T}^{k}{y}_{0},{T}^{k}{y}_{r+1})\vert \leq (p+1)\Vert {\psi }_{r+1}\Vert$ we obtain that

$$\begin{eqnarray*} &&\frac{{\mathcal{P}}_{r}^{(p)}[a]}{{\mathcal{P}}_{r+1}^{(p)}[a]}={\mathrm{e}}^{\pm 2(p+1)\Vert {\psi }_{r+1}\Vert }. \end{eqnarray*}$$

Finally, putting together these previous bounds yields

$$\begin{eqnarray*} \frac{{\mu }_{r}[a]}{{\mu }_{r+1}[a]}&=&\exp \left (\pm 2(p+1)\Vert {\psi }_{r+1}\Vert \right )\cdot \exp \left (\pm 2 4(r+1){C}_{r+1}{\mathrm{e}}^{2{C}_{r+1}}{\eta }^{(p-q)/(r+1)-2}\right )\\ &=&\exp \left (\pm \bigl (2(p+1)\Vert {\psi }_{r+1}\Vert +2 4(r+1){C}_{\infty }{\mathrm{e}}^{2{C}_{\infty }}{\eta }^{(p-n-r-1)/(r+1)-2}\bigr )\right ), \end{eqnarray*}$$

where ${C}_{\infty }={\mathop{\sum }\nolimits }_{k=1}^{\infty }k\Vert {\psi }_{k}\Vert$. Let p:=p(s), then for any word a∈Aⁿ and any r' > r∈N₀, one has that

$$\begin{eqnarray*} &&\frac{{\mu }_{r}[a]}{{\mu }_{{r}^{\prime}}[a]}=\exp \left (\pm \left (2\sum _{s=r}^{{r}^{\prime}}p(s)\Vert {\psi }_{s}\Vert +2 4 R\sum _{s=r}^{{r}^{\prime}}s{\eta }^{(p(s)/s)}\right )\right ), \end{eqnarray*}$$

where R:=C_∞e^2C_∞. Since η < 1 and assuming condition 1, the right-hand side of the previous expression tends to zero as r diverges. Thus, the limit μ*[a]:=lim_r→∞μ_r[a], exists for any word a∈Aⁿ, proving the weak-∗ convergence. Moreover, one has that

$$\begin{eqnarray} &&\frac{{\mu }_{r}[a]}{{\mu }^{\ast }[a]}=\exp \left (\pm \left (2\mathop{\sum }\limits _{s\geq r}p(s)\Vert {\psi }_{s}\Vert +2 4 R\mathop{\sum }\limits _{s\geq r}s{\eta }^{(p(s)/s)}\right )\right ), \end{eqnarray} \tag{ 5 }$$

for every r∈N₀ and any word a∈Aⁿ. So one also has that μ_r[a]/μ*[a] = e^±nε_r, where n is the length of the word a, and ε_r is given by

$$\begin{eqnarray*} &&{\varepsilon }_{r}:= 2\mathop{\sum }\limits _{s\geq r}p(s)\Vert {\psi }_{s}\Vert +2 4 R\mathop{\sum }\limits _{s\geq r}s{\eta }^{p(s)/s}, \end{eqnarray*}$$

which proves the projective convergence of the Markov approximations μ_r to μ*. The limiting measure μ* turns out to be absolutely continuous to μ since we use the elementary Gibbs measures to approximate the Markov ones. The proof of this statement is found in [6, section 6.3.3.]. That concludes the proof. □

4.1.1. Proof of Lemma 4.1.

Here we use that every Markov measure can be seen as a limit of measures with support on the periodic points as was proved in [6, Appendix 6.3.1.]. From the calculations in that paper, one has explicit bounds for the Markov measure μ_r in terms of the finite-range periodic approximation ${\mathcal{P}}_{r}^{(p)}$ as follows:

$$\begin{eqnarray*} &&{\mu }_{r}[a]={\mathcal{P}}_{r}^{(p)}[a]\exp \left (\pm \frac{r{D}_{0}}{1-\eta }{\eta }^{(p-n)/r-2}\right ), \end{eqnarray*}$$

where $\eta := \eta ({M}_{r}^{r})$, is the Birkhoff coefficient of the matrix ${M}_{r}^{r}$ and D₀ is a constant bounded by

$$\begin{eqnarray*} &&{D}_{0}\leq 2\mathop{\max }\limits _{b,c,{c}^{\prime}\in {A}^{r}}\log \left (\frac{M_{r}^{r}(c,b){M}_{r}^{r+1}({c}^{\prime},b)}{M_{r}^{r+1}(c,b){M}_{r}^{r}({c}^{\prime},b)}\right )=\frac{{\mathrm{e}}^{{U}_{r,r}(c b)}{\mathrm{e}}^{{U}_{r+1,r}({c}^{\prime}b)}}{{\mathrm{e}}^{{U}_{r+1,r}(c b)}{\mathrm{e}}^{{U}_{r,r}({c}^{\prime}b)}}. \end{eqnarray*}$$

Using that

$$\begin{eqnarray*} &&\vert {U}_{r,r}(b{c}^{\prime})-{U}_{r+1,r}(b{c}^{\prime})\vert \leq 3\sum _{k=1}^{r}k\Vert {\psi }_{k}\Vert , \end{eqnarray*}$$

for b∈A^r and c'∈A, one gets

$$\begin{eqnarray*} &&{D}_{0}\leq 2{C}_{r}, \end{eqnarray*}$$

where ${C}_{r}:= {\mathop{\sum }\nolimits }_{k=1}^{r}k\Vert {\psi }_{k}\Vert$. And so, one has

$$\begin{eqnarray*} &&{\mu }_{r}[a]={\mathcal{P}}_{r}^{(p)}[a]\cdot \exp \left (\pm 1 2 r{C}_{r}{\mathrm{e}}^{2{C}_{r}}\times {\eta }^{(p-n)/r-2}\right ), \end{eqnarray*}$$

which proves the lemma for $\eta =1-{\mathrm{e}}^{-2{\mathop{\sum }\nolimits }_{k=1}^{\infty }k\Vert {\psi }_{k}\Vert }$ and ${C}_{r}={\mathop{\sum }\nolimits }_{k=1}^{r}k\Vert {\psi }_{k}\Vert$. □

Here we mimic the proof of Theorem 3.3 in [5]. As before we make use of the periodic approximation of the limiting measure through the following lemma.

Lemma 4.2 Let ${\mathcal{P}}_{r}^{(p)}$ be the r-range approximation of the periodic measure ^(p). Let s,r∈N₀. For any a,b∈A^r one has that

$$\begin{eqnarray} &&{\mathcal{P}}_{r}^{(p)}([a]\cap {T}^{-s}[b])={\mathcal{P}}_{r}^{(p)}[a]{\mathcal{P}}_{r}^{(p)}[b]\times \exp \left (\pm 4\frac{r D}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right ), \end{eqnarray} \tag{ 6 }$$

for a constant D > 0 and where η is the same constant as in Lemma 4.1.

From inequality (5) applied to the limiting measure one can easily obtain that

$$\begin{eqnarray*} &&\fl \vert \mu ([a]\cap {T}^{-s}[b])-\mu [a]\mu [b]\vert \leq \vert {\mathcal{P}}_{r}^{(p)}([a]\cap {T}^{-s}[b])-{\mathcal{P}}_{r}^{(p)}[a]{\mathcal{P}}_{r}^{(p)}[b]\vert \\ &&\times ~\exp \left (2\Bigl [2\mathop{\sum }\limits _{k\geq r}p(k)\Vert {\psi }_{k}\Vert +2 4 R\mathop{\sum }\limits _{k\geq r}k{\eta }^{p(k)/k}\Bigr ]\right ). \end{eqnarray*}$$

Using the inequality (6) from our previous lemma yields

$$\begin{eqnarray*} &&\fl \left \vert \frac{\mu ([a]\cap {T}^{-s}[b])}{\mu [a]\mu [b]}-1\right \vert \leq \left \vert \exp \left (4 r\frac{D}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right )-1\right \vert \\ &&\times ~\exp \left (2\left [2\mathop{\sum }\limits _{k\geq r}p(k)\Vert {\psi }_{k}\Vert +2 4 R\mathop{\sum }\limits _{k\geq r}k{\eta }^{p(k)/k}\right ]\right ). \end{eqnarray*}$$

By the hypothesis made on the interaction, the second factor in the right-hand side of the last inequality can be bounded by above by a constant C > 1 for every r ≥ 1. The larger r we take, the smaller the sufficient upper bound C becomes.

Next, let us consider r = r(s) such that r(s) = o(s). Then there exists a constant 0 < ξ < 1 such that for every pair a,b∈A^r there exists a s*∈N₀ for which

$$\begin{eqnarray*} &&\left \vert \frac{\mu ([a]\cap {T}^{-s}[b])}{\mu [a]\mu [b]}-1\right \vert \leq C s{\eta }^{{s}^{\xi }}, \end{eqnarray*}$$

for every s ≥ s*. Now, by introducing an adequate positive constant c we change the base and it is possible to rewrite the previous inequality into the form

$$\begin{eqnarray*} &&\left \vert \frac{\mu ([a]\cap {T}^{-s}[b])}{\mu [a]\mu [b]}-1\right \vert \leq C s{\mathrm{e}}^{-c{s}^{\xi }}, \end{eqnarray*}$$

again for every s ≥ s*. And that finishes the proof of the theorem. □

4.2.1. Proof of Lemma 4.2.

First, let r∈N, p > 2r and let M_r be the transition matrix given by (4). Consider any a∈A^r. We have

$$\begin{eqnarray*} &&{\mathcal{P}}_{r}^{(p)}[a]=\frac{\mathop{\sum }_{x\in {\mathrm{Per}}_{p}(X)\cap [a]}{\mathrm{e}}^{{U}_{p,r}(x)}}{\mathop{\sum }_{x\in {\mathrm{Per}}_{p}(X)}{\mathrm{e}}^{{U}_{p,r}(x)}}=\frac{{e}_{a}^{\dagger }{M}_{r}^{p}{e}_{a}}{\mathop{\sum }_{x\in {A}^{r}}\mathrm{e}_{x}^{\dagger }{M}_{r}^{p}{e}_{x}}, \end{eqnarray*}$$

where e_x is a vector with an entry equal to 1 at position x and 0 elsewhere. For any s∈N₀, it is easy to see that

$$\begin{eqnarray*} &&{\mathcal{P}}_{r}^{(p)}([a]\cap {T}^{-s}[b])=\mathop{\sum }\limits _{w\in {A}^{s}}\mathcal{P}_{r}^{(p)}[a w b]. \end{eqnarray*}$$

Next, choose an arbitrary but fixed w∈A^s. We are able to write ${\mathcal{P}}_{r}^{(p)}[a w b]$ in terms of the transition matrix M_r, as follows:

$$\begin{eqnarray*} &&{\mathcal{P}}_{r}^{(p)}[a w b]=\frac{{e}_{a}^{\dagger }{M}_{r}^{p-a}{e}_{w b}{e}_{w b}^{\dagger }{M}_{r}^{s}{e}_{a}}{\mathop{\sum }_{a}\mathop{\sum }_{w b}e_{a}^{\dagger }M_{r}^{p-s}{e}_{w b}e_{w b}^{\dagger }M_{r}^{s}{e}_{a}}. \end{eqnarray*}$$

Here we use the enhanced Perron–Frobenius theorem [6, Corollary 2.16.], yielding

$$\begin{eqnarray*} &&\fl {\mathcal{P}}_{r}^{(p)}[a w b]=\frac{{e}_{a}^{\dagger }({L}^{\dagger }{e}_{w b}){\rho }^{p-s}R\cdot {e}_{w b}^{\dagger }({L}^{\dagger }{e}_{a}){\rho }^{s}R}{\mathop{\sum }_{a}\mathop{\sum }_{w b}e_{a}^{\dagger }({L}^{\dagger }{e}_{w b}){\rho }^{p-s}R\cdot e_{w b}^{\dagger }({L}^{\dagger }{e}_{a}){\rho }^{s}R}\\ &&\times ~\exp \left (\pm 2\frac{r \delta ({e}_{w b},F{e}_{w b})}{1-\eta }{\eta }^{\lfloor (p-s)/r\rfloor }\right )\cdot \exp \left (\pm 2\frac{r \delta ({e}_{a},F{e}_{a})}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right ), \end{eqnarray*}$$

where F is the contraction defined on the simplex and δ is a distance defined on the simplex—both are explicitly defined in [5, Appendix 1.] and [6, Appendix 6.1.]. Since L^†e_x = L(x) and ${e}_{x}^{\dagger }R=R(x)$ one has that

$$\begin{eqnarray*} &&\fl {\mathcal{P}}_{r}^{(p)}[a w b]=\frac{L(w b)R(a)\cdot L(a)R(w b)}{\mathop{\sum }_{a}\mathop{\sum }_{w b}L(w b)R(a)\cdot L(a)R(w b)}\\ &&\times ~\exp \left (\pm 2\frac{r \delta ({e}_{w b},F{e}_{w b})}{1-\eta }{\eta }^{\lfloor (p-s)/r\rfloor }\right )\cdot \exp \left (\pm 2\frac{r \delta ({e}_{a},F{e}_{a})}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right )\\ &&=~{\mathcal{P}}_{r}^{(p)}(a){\mathcal{P}}_{r}^{(p)}(w b)\\ &&\times ~\exp \left (\pm 2\frac{r \delta ({e}_{w b},F{e}_{w b})}{1-\eta }{\eta }^{\lfloor (p-s)/r\rfloor }\right )\cdot \exp \left (\pm 2\frac{r \delta ({e}_{a},F{e}_{a})}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right ), \end{eqnarray*}$$

where in the last inequality we have used the fact that is a probability measure. Finally, one obtains that

$$\begin{eqnarray*} &&\fl {\mathcal{P}}_{r}^{(p)}([a]\cap {T}^{-s}[b])=\mathop{\sum }\limits _{w\in {A}^{s}}\mathcal{P}_{r}^{(p)}(a){\mathcal{P}}_{r}^{(p)}(w b)\\ &&\times ~\exp \left (\pm 2\frac{r \delta ({e}_{w b},F{e}_{w b})}{1-\eta }{\eta }^{\lfloor (p-s)/r\rfloor }\right )\cdot \exp \left (\pm 2\frac{r \delta ({e}_{a},F{e}_{a})}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right )\\ &&=~~{\mathcal{P}}_{r}^{(p)}(a){\mathcal{P}}_{r}^{(p)}(b)\times \exp \left (\pm 4\frac{r D}{1-\eta }{\eta }^{\lfloor s/r\rfloor }\right ), \end{eqnarray*}$$

for some constant D > 0. Recall that in our particular case the Birkhoff coefficient η is giving by $\eta =1-{\mathrm{e}}^{-2{\mathop{\sum }\nolimits }_{k=1}^{\infty }k\Vert {\psi }_{k}\Vert }$. □

First notice that given r > 0 and given a∈^k with k > 0 we have that for every x,y∈[a]

$$\begin{eqnarray*} &&\vert u(x)-{u}_{r}(y)\vert =\sum _{n=r}^{\infty }{\psi }_{n}({x}_{0},{x}_{n}) \end{eqnarray*}$$

then we have that, for every k > 0 the sum difference |U_k(x) − U_k,r(y)| can be bounded by

$$\begin{eqnarray*} &&\vert {U}_{k}(x)-{U}_{k,r}(y)\vert =\sum _{n=r}^{\infty }\sum _{j=0}^{k-1}{\psi }_{n}({x}_{j},{x}_{j+n})\leq k\sum _{n=r}^{\infty }\Vert {\psi }_{n}\Vert . \end{eqnarray*}$$

Since ψ has summable variation it is clear that the last sum is finite and goes to zero as r → ∞. Moreover, for polynomially decaying interaction, such can be bounded by

$$\begin{eqnarray*} &&\sum _{n=r}^{\infty }\Vert {\psi }_{n}\Vert \leq {C}^{\prime}{r}^{-\epsilon } \end{eqnarray*}$$

for some  > 0. The last constant depends on how the pair-wise interaction ψ_k decays with the 'distance' k. From the above we can easily see that

$$\begin{eqnarray*} &&\exp \left \{ {U}_{k}({a}^{\ast })\right \} =\exp \left \{ {U}_{k,r}({a}^{\ast })\right \} \exp \left (\pm k{C}^{\prime}{r}^{-\epsilon }\right ). \end{eqnarray*}$$

This result lets us bound the difference of topological pressures as follows:

$$\begin{eqnarray*} &&0\leq \vert P(u,X)-P({u}_{r},X)\vert \leq \frac{1}{k}\log \biggl (\frac{\mathop{\sum }_{a\in {\mathrm{Per}}_{k}}\exp \left \{ {U}_{k}({a}^{\ast })\right \} }{\mathop{\sum }_{b\in {\mathrm{Per}}_{k}}\exp \left \{ {U}_{k,r}({b}^{\ast })\right \} }\biggr )\leq {C}^{\prime}{r}^{-\epsilon }, \end{eqnarray*}$$

where a* (b* respectively) is some point X belonging to [a] ([b] respectively).

Now, since both μ_r and μ satisfy the variational principle [3] we have that

$$\begin{eqnarray*} &&P(u,X)=\int _{X}u\mathrm{d}\mu +h(\mu )\tqs \mathrm{and}\tqs P({u}_{r},X)=\int _{X}{u}_{r}\mathrm{d}{\mu }_{r}+h({\mu }_{r}), \end{eqnarray*}$$

which allows us to write

$$\begin{eqnarray} &&\vert h(\mu )-h({\mu }_{r})\vert \leq \vert P(u,X)-P({u}_{r},X)\vert +\vert \int _{X}u\mathrm{d}\mu -\int _{X}{u}_{r}\mathrm{d}{\mu }_{r}\vert . \end{eqnarray} \tag{ 7 }$$

Now, by (5) we know that

$$\begin{eqnarray*} &&\mu ={\mu }_{r}\exp (\pm {\varepsilon }_{r}), \end{eqnarray*}$$

which allows us to write

$$\begin{eqnarray*} &&\Biggl \vert \int _{X}u\mathrm{d}\mu -\int _{X}{u}_{r}\mathrm{d}{\mu }_{r}\Biggr \vert \leq \left (1-\exp (-{\varepsilon }_{r})\right )\Vert u\Vert . \end{eqnarray*}$$

It is clear that the right-hand side of the above equation vanishes exponentially, which allows us to conclude that the left-hand side of inequality (7) is bounded by Cr⁻ choosing an appropriate constant C > 0,

$$\begin{eqnarray*} &&\vert h(\mu )-h({\mu }_{r})\vert \leq C{r}^{-\epsilon }, \end{eqnarray*}$$

which proves the theorem. □