2.1. Finite volume measure

The space of configurations is the set of locally finite subsets of $\mathbb{R}^d$ ,

\begin{equation*}\Omega = \{\omega\subset\mathbb{R}^d\,{:}\,\vert\omega\cap\Delta\vert < \infty,\; \text{for all } \Delta \subset\mathbb{R}^d,\, \text{bounded} \},\end{equation*}

where $\vert \cdot \vert$ is the cardinal. We denote $\omega \cap \Delta$ by $\omega_{\Delta}$ , the union $\omega\cup\omega'$ of two configurations by $\omega\omega'$ , the space of finite configurations by $\Omega_{\rm f}$ and the space of configurations in $\Lambda\subset \mathbb{R}^d$ by $\Omega_{\Lambda}$ .

Our space is equipped with the sigma field $\mathcal{F}$ generated by the counting functions $N^{}_{\Delta}\,{:}\,\omega \mapsto \vert \omega_{\Delta} \vert$ for all $\Delta \subset \mathbb{R}^d$ bounded. In our setting, a point process is simply a probability measure on $(\Omega, \mathcal{F})$ . Note that with this definition we identify a point process with its distribution. We say that the process has a finite intensity if, for all bounded subsets $\Delta$ , the expectation $\mathbb{E}_{P}[\vert \omega_{\Delta} \vert]$ is finite. If we denote this expectation by $\mu(\Delta)$ then $\mu$ is a sigma-finite measure on $\mathbb{R}^d$ called the intensity measure. When $\mu = i(P) \lambda^d$ , where $\lambda^d$ is the Lebesgue measure on $\mathbb{R}^d$ and $i(P) \geq 0$ , we simply say that the point process has intensity i(P). We also introduce

\begin{equation*}\xi(P) = \sup_{\substack{\Lambda \subset \mathbb{R}^d \\ 0 < \lambda^d(\Lambda) < +\infty}} \frac{\mathbb{E}^P[\vert \omega_{\Lambda}^{}\vert]}{\lambda^d(\Lambda)} ,\end{equation*}

and we say that a probability measure P has a bounded intensity if $\xi(P) < +\infty$ . Obviously, if P has a finite intensity i(P) then $\xi(P) = i(P)$ .

A point process P is stationary if, for all $u\in\mathbb{R}^d$ , $P = P\circ\tau^{-1}_{u}$ , where $\tau_u$ is the translation of vector u. If a stationary point process has a finite intensity then its intensity measure is proportional to the Lebesgue measure and has the form $\mu = i(P) \lambda^d$ .

The most popular point processes are the Poisson point processes. We consider here only the homogeneous (or stationary) case where the intensity has the form $\mu = z \lambda^d$ . The process is denoted $\pi^{z}$ , or simply $\pi$ if $z = 1$ . Recall briefly that $\pi^{z}$ is the only point process in $\mathbb{R}^d$ with intensity $\mu = i(P) \lambda^d$ such that any two disjoint regions of space are independent under $\pi^{z}$ . See the recent book [Reference Last and Penrose12] on the subject.

Let us now define the interaction between the points. We need to introduce an energy function.

A crucial assumption is the stability of the energy function.

Definition 2. An energy function is said to be stable if there exists a constant $B \geq 0$ such that, for all $\omega \in \Omega_{\rm f}$ ,

\begin{equation*}H(\omega) \geq -B \vert \omega \vert.\end{equation*}

The above assumption (denoted [Stable]) is standard and has been treated deeply in the literature (see, for instance, [Reference Ruelle14, Section 3.2]).

Now we can define the Gibbs point processes in finite volume.

Definition 3. Let $\Lambda$ be a bounded subset in $\mathbb{R}^d$ . The Gibbs point process on $\Lambda$ for the stable energy function H is the probability measure on $\Omega_{\Lambda}$ defined by

\begin{equation*}P_{\Lambda}({\rm d}\omega) = \frac{1}{Z_{\Lambda}} {\rm e}^{-H(\omega)} \pi_{\Lambda}({\rm d}\omega) ,\end{equation*}

with the normalization constant $Z_{\Lambda} = \int {\rm e}^{- H(\omega)} \pi_{\Lambda}({\rm d}\omega)$ called the partition function.

We can check with the properties of H that $P_{\Lambda}$ is well defined (i.e. $0 < Z_{\Lambda} < +\infty$ ). In comparison with the standard formalism of Gibbs measures in statistical physics (see [Reference Ruelle14], for instance), the activity and inverse temperature parameters are included in the function H here.

2.2. Infinite volume measure

Let us turn now to the definition of Gibbs point processes in the infinite volume regime. For the following, we choose the convention $\infty - \infty = 0$ . We need to introduce the local energy, which is given by $H_{\Delta}(\omega) = H(\omega) - H(\omega_{\Delta^{\rm c}})$ for a finite configuration $\omega\in\Omega_{\rm f}$ . It represents the contribution of energy coming from $\omega_{\Delta}$ in $\omega$ (the difference of energies within and without $\omega_{\Delta}$ ). We need to extend this definition to an infinite configuration. If $(\Delta_l)_{l\geq 0}$ is an increasing sequence of subsets in $\mathbb{R}^d$ with $\Delta_l \uparrow \mathbb{R}^d$ , we expect that the limit $\displaystyle \lim_{l\rightarrow +\infty} H^{l}_{\Delta}(\omega)$ exists, where

\begin{equation*}H^{l}_{\Delta}(\omega) = H\big(\omega^{}_{\Delta_l}\big) - H\big(\omega^{}_{\Delta_l \setminus \Delta}\big) \in \mathbb{R}\cup\{+\infty\}.\end{equation*}

In particular, the difference $H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega)$ should go to zero as l goes to infinity. Our main assumption is a control of the expectation of this difference for point processes with bounded intensities.

Definition 4. An energy function is said to be intensity regular if, for all bounded subsets $\Delta$ of $\mathbb{R}^d$ , we can find an increasing sequence of subsets $(\Delta_l)_{l\geq 0}$ with $\Delta_l \uparrow \mathbb{R}^d$ and $\Delta \subset \Delta_0$ such that, for all configurations $\omega$ ,

\begin{equation*}\left| H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) \right| \leq \vert \omega_{\Delta} \vert G_{\Delta}^l(\omega_{\Delta^{\rm c}}),\end{equation*}

where $G^{l}_{\Delta}$ is a non-negative measurable function on $\Omega_{\Delta^{\rm c}}$ such that, for any probability measure P with bounded intensity, we have

\begin{equation*}\mathbb{E}_P\left[G_{\Delta}^l(\omega_{\Delta^{\rm c}}) \right] \leq \alpha_l\, a(\xi(P)),\end{equation*}

with $a\,{:}\,\mathbb{R}_+ \rightarrow \mathbb{R}_+$ an increasing function and $(\alpha_l)_{l\geq 0}$ a positive summable sequence (i.e. $\sum_{l\ge 0} \alpha_l<+\infty$ ). Let P be a probability measure with bounded intensity. From the above assumption (denoted [Intensity Regular]), we have

\begin{equation*}\sum_{l=0}^{+\infty}\left| H_{\Delta}^{l+1}(\omega) - H_{\Delta}^l(\omega) \right| \leq \vert \omega_{\Delta} \vert \sum_{l=0}^{+\infty} G^l_{\Delta}(\omega_{\Delta^{\rm c}}),\end{equation*}

and we can control the expectation of the second part of the right-hand side:

\begin{align*}\mathbb{E}_P\left[\sum_{l=0}^{+\infty} G^l_{\Delta}(\omega_{\Delta^{\rm c}}) \right] \leq a(\xi(P)) \sum_{l=0}^{+\infty} \alpha_l < + \infty.\end{align*}

Then, the local energy is correctly defined for P-almost every configuration $\omega$ by

\begin{equation*}H_{\Delta}(\omega) = H_{\Delta}^{0}(\omega) + \sum_{l=0}^{+\infty} \left[H_{\Delta}^{l+1}(\omega) - H_{\Delta}^l(\omega) \right].\end{equation*}

In addition, if we introduce the function $C^l_{\Delta}$ defined on $\Omega_{\Delta^{\rm c}}$ by

\begin{equation*}C^l_{\Delta}(\omega_{\Delta^{\rm c}}) = \sum_{j=l}^{+\infty} G^j_{\Delta}(\omega_{\Delta^{\rm c}}),\end{equation*}

then for all configurations $\omega \in \Omega$ we have the following approximation result:

(1) \begin{equation}\left| H_{\Delta}(\omega) - H_{\Delta}^l(\omega) \right| \leq \vert \omega_{\Delta} \vert C_{\Delta}^l(\omega_{\Delta^{\rm c}}),\end{equation}

with $\displaystyle \mathbb{E}_P[C_{\Delta}^l(\omega_{\Delta^{\rm c}})] \leq a(\xi(P)) \sum_{j=l}^{\infty}\alpha_j$ .

We are now able to give the definition of infinite-volume Gibbs point processes.

Definition 5. A probability measure P on $\Omega$ with bounded intensity is a Gibbs point process for an energy function H, satisfying assumptions [Stable] and [Intensity Regular], if, for all bounded subsets $\Delta$ and all bounded measurable functions f, we have

(2) \begin{equation}\int f(\omega) P({\rm d}\omega)=\int \int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}) \frac{1}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})} {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) P({\rm d}\omega),\end{equation}

with the normalization constant $Z_{\Delta}(\omega^{}_{\Delta^{\rm c}}) = \int {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta})$ .

The equations (2) for all $\Delta$ and f are the DLR equations. To be correctly defined we need to check that $0 < Z_{\Delta}(\omega^{}_{\Delta^{\rm c}}) < +\infty$ for P-almost all configurations $\omega$ . A lower bound can easily be obtained with

\begin{equation*}Z_{\Delta}(\omega^{}_{\Delta^{\rm c}}) \geq {\rm e}^{-H_{\Delta}(\omega_{\Delta^{\rm c}})}\pi_{\Delta}(\emptyset) = {\rm e}^{-\lambda^d(\Delta)} > 0.\end{equation*}

For an upper bound, we use assumptions [Stable] and [Intensity Regular]. From inequality (1), and assuming that $H(\omega_{\Delta_0\setminus\Delta})<+\infty$ , we have

\begin{align*}H_{\Delta}(\omega) & \geq H_{\Delta}^{0}(\omega) - \vert \omega_{\Delta} \vert C_{\Delta}^{0}(\omega_{\Delta^{\rm c}}) \\[3pt]& = H(\omega_{\Delta_0}) - H(\omega_{\Delta_0 \setminus \Delta}) - \vert \omega_{\Delta} \vert C_{\Delta}^{0}(\omega_{\Delta^{\rm c}}) \\[3pt]& \geq -(B + C_{\Delta}^{0}(\omega_{\Delta^{\rm c}})) \vert \omega_{\Delta} \vert - B \vert \omega_{\Delta_0 \setminus \Delta}\vert - H(\omega_{\Delta_0 \setminus \Delta}).\end{align*}

Then, we obtain

\begin{align*}Z_{\Delta}(\omega^{}_{\Delta^{\rm c}}) & \leq \exp\left(B \vert \omega_{\Delta_0 \setminus \Delta}\vert + H(\omega_{\Delta_0 \setminus \Delta})\right)\int {\rm e}^{ \big(C_{\Delta}^{0}(\omega_{\Delta^{\rm c}}) + B\big) \vert \omega^{\prime}_{\Delta} \vert} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \\ & = \exp\left(B \vert \omega_{\Delta_0 \setminus \Delta}\vert + H(\omega_{\Delta_0 \setminus \Delta})\right) \sum_{n=0}^{+\infty} \frac{\big(\lambda^d(\Delta){\rm e}^{ \big(C_{\Delta}^{0}(\omega_{\Delta^{\rm c}}) + B\big)}\big)^n}{n!} {\rm e}^{-\lambda^d(\Delta)} \\ & = \exp\left(B \vert \omega_{\Delta_0 \setminus \Delta}\vert + H(\omega_{\Delta_0 \setminus \Delta}) +\lambda^d(\Delta)\big({\rm e}^{ \big(C_{\Delta}^{0}(\omega_{\Delta^{\rm c}}) + B\big)} - 1\big)\right) \\ & < +\infty.\end{align*}

If $H(\omega_{\Delta_0\setminus\Delta}) = +\infty $ then, by the hereditary property, for all $l \geq 0$ we have $H(\omega_{\Delta_l\setminus\Delta}) =\break +\infty$ and $H^l_\Delta(\omega) = 0$ (according to our convention $\infty - \infty = 0$ given above), which gives $H_\Delta(\omega) = 0$ and $Z_\Delta(\omega_{\Delta^{\rm c}}) < +\infty$ .

Our main result is the following theorem, which is proved in Section 4.

3.1. Finite-range interaction

An energy function H has a finite range if there exists $R>0$ such that, for all finite configurations $\omega \in \Omega_{\rm f}$ and bounded subset $\Delta$ , $H_{\Delta}(\omega) = H_{\Delta}\left(\omega_{\Delta\oplus B(0,R)}\right)$ . It is easy to see that a finite-range energy function verifies assumption [Intensity Regular], with any increasing sequence of subsets $(\Delta_l)_{l\geq 0}$ such that $\Delta_0 = \Delta\oplus B(0,R)$ . Therefore, in this setting of finite-range interaction, only the assumption [Stable] is required to ensure the existence of Gibbs point processes.

This result was proved previously in [Reference Dereudre, Drouilhet and Georgii5] (see also [Reference Dereudre4] for a simpler and pedagogical proof).

3.2. Pairwise interaction

An energy function H is pairwise if there exists a symmetric function $\phi\,{:}\,\mathbb{R}^d \mapsto \mathbb{R} \cup\{+\infty\}$ , called a potential, such that

\begin{equation*}H(\omega)= \sum_{\{x,y\} \subset \omega} \phi(x - y).\end{equation*}

We do not assume the finite-range property and so the support of $\phi$ can be unbounded. However, we still need an assumption on the long-range behavior of $\phi$ . The potential $\phi$ is regular if there exists a decreasing function $\psi \,{:}\,\mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that $\int_A^{+\infty} \psi(r)r^{d-1}\,{\rm d} r < +\infty$ for some $A>0$ and $\vert \phi(x) \vert \leq \psi(\Vert x\Vert)$ for all $x\in \mathbb{R}^d$ . Let us show that this energy function satisfies the assumption [Intensity Regular].

Let $\Delta$ be a bounded subset of $\mathbb{R}^d$ . We choose $z\in\mathbb{R}^d$ and $\rho>0$ such that $\Delta\subset B(z,\rho)$ , and we introduce $\Delta_l = B(z,\rho+A+l)$ . We have

\begin{equation*} H^l_{\Delta}(\omega) = \sum_{\substack{\{x,y\}\subset\omega\cap\Delta_l\\ \{x,y\}\cap\Delta \neq \emptyset}} \phi(x- y),\end{equation*}

which gives

\begin{equation*} H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) = \sum_{x\in\Delta}\, \sum_{y\in\Delta_{l+1}\setminus\Delta_l} \phi( x-y).\end{equation*}

By definition, if $x\in\Delta$ and $y\in\Delta_{l+1}\setminus\Delta_l$ then $\Vert x - y \Vert \geq A + l$ and $\vert\phi(x-y)\vert \leq \psi(A\,{+}\,l)$ . Introducing the function $G^l_\Delta(\omega_{\Delta^{\rm c}}) = \vert \omega_{\Delta_{l+1}\setminus{\Delta_l}} \vert \psi(A+l)$ , we have $\vert H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) \vert \leq \vert \omega_{\Delta} \vert G^l_{\Delta}(\omega_{\Delta^{\rm c}})$ and, for all P with bounded intensity,

\begin{align*} \mathbb{E}_P\left[G^l_{\Delta}(\omega_{\Delta^{\rm c}})\right] &\leq \xi(P)\lambda^d(\Delta_{l+1}\setminus\Delta_l) \psi(A+l) \\ & \leq \xi(P) c_d\left((\rho + A + l + 1)^d - (\rho + A + l)^d\right)\psi(A+l)\\ & \leq \xi(P) c_d d(\rho+A+l+1)^{d-1} \psi(A+l) .\end{align*}

Since $\int_A^{+\infty}\psi(r)r^{d-1}\,{\rm d} r<+\infty$ and $\psi$ is decreasing, we have that

\begin{align*} \sum_{l\geq0}(\rho+A+l+1)^{d-1} \psi(A+l) < +\infty.\end{align*}

Hence, the energy H satisfies the assumption [Intensity Regular] with $a = id$ and $\alpha_l = c_d d(\rho+A+l+1)^{d-1}\psi(A+l)$ .

The stability assumption for a pairwise potential is delicate and was investigated a long time ago. We refer to [Reference Ruelle14] for several results. We mention that a non-negative potential which is equal to zero in the neighborhood of zero and positive outside this neighborhood is a simple example of a stable but not a superstable potential.

Note that the sum of two energy functions satisfying assumption [Intensity Regular] also satisfies the assumption [Intensity Regular]. The following corollary follows.

3.3. Cloud interaction

In this last example, we provide an energy function which is infinite range and not reducible, at any scale, to a pairwise interaction. It is a multibody interaction between a germ–grain interaction (see, for instance, the Quermass model [Reference Dereudre3] or the Widom–Rowlinson interaction [Reference Widom and Rowlinson16]) and a pairwise interaction. We call it a cloud interaction because each point of the configuration is diluted in a cloud around itself and the pairwise interaction is integrated on this cloud. Precisely, for any finite configuration $\omega$ ,

\begin{equation*} H(\omega) = \sum_{x\in\omega}\int_{L^R(\omega)} \phi(x - y) \,{\rm d} y ,\end{equation*}

where $ L^R(\omega) = \bigcup_{x\in\omega} B(x,R)$ is the cloud produced by the configuration $\omega$ ( $R>0$ is a fixed parameter) and $\phi\,{:}\,\mathbb{R}^d \mapsto \mathbb{R} \cup\{\infty\}$ is a symmetric function. This energy function can be viewed as an approximation of the pairwise interaction introduced above. Indeed, $H(\omega)/R^d$ tends to the pairwise interaction function (times a multiplicative constant) when R goes to zero.

We assume that there exists a decreasing function $\psi\,{:}\,\mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that, for all $x\in\mathbb{R}^d$ , $\vert\phi(x)\vert \leq \psi(\Vert x \Vert)$ and $\int \psi(r) r^{d-1} \,{\rm d} r<+\infty$ . This ensures that the energy function is well defined and satisfies the assumption [Stable] since

\begin{equation*}|H(\omega)| \le |\omega| \int \vert\psi(r)\vert r^{d-1} \,{\rm d} r.\end{equation*}

Note that H is not superstable. Indeed, this would require that for all $\omega\in\Omega_{\Lambda}$ , $H(\omega) \geq -B\vert\omega\vert + A\vert \omega \vert^2 / \lambda^d(\Lambda)$ with $B\geq 0$ and $A > 0$ , which is impossible with the previous upper bound of H.

Concerning assumption [Intensity Regular], for $\Delta$ a bounded subset of $\mathbb{R}^d$ , the local energy is given by

\begin{align*} H^l_{\Delta}(\omega) & = H(\omega_{\Delta_l}) - H(\omega_{\Delta_l\setminus\Delta}) \\ & = \sum_{x\in\omega_{\Delta_l}}\int_{L^R(\omega_{\Delta_l})} \phi(\Vert x - y \Vert) \,{\rm d} y - \sum_{x\in\omega_{\Delta_l\setminus\Delta}}\int_{L^R(\omega_{\Delta_l\setminus\Delta})} \phi(\Vert x - y \Vert) \,{\rm d} y \\ & = \sum_{x\in\omega_{\Delta_l\setminus\Delta}}\int_{L^R(\omega_{\Delta_l})\setminus L^R(\omega_{\Delta_l\setminus\Delta})} \phi(\Vert x - y \Vert) \,{\rm d} y + \sum_{x\in\omega_{\Delta}} \int_{L^R(\omega_{\Delta_l})} \phi(\Vert x - y \Vert) \,{\rm d} y.\end{align*}

We can find $z\in\mathbb{R}^d$ and $\rho >0$ such that $\Delta \subset B(z,\rho)$ , and, for the following, we choose $\Delta_0= B(z, \rho+ 2R)$ , which implies that

\begin{equation*} L^R(\omega_{\Delta_l})\setminus L^R(\omega_{\Delta_l\setminus\Delta}) = L^R(\omega_{\Delta_0})\setminus L^R(\omega_{\Delta_0\setminus\Delta}) \overset{\textrm{def}}{=} L^R_{\Delta}(\omega).\end{equation*}

Using this notation we have

\begin{equation*} H^l_{\Delta}(\omega) = \sum_{x\in\omega_{\Delta_l\setminus\Delta}}\int_{L^R_{\Delta}(\omega)} \phi(\Vert x - y \Vert) \,{\rm d} y + \sum_{x\in\omega_{\Delta}} \int_{L^R(\omega_{\Delta_l})} \phi(\Vert x - y \Vert) \,{\rm d} y.\end{equation*}

The first term corresponds to the interaction of the points outside of $\Delta$ with the cloud created by the points in $\Delta$ , and the second corresponds to the interaction of the points in $\Delta$ with the full cloud. We can compute the cost of adding a shell:

\begin{align*} H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) &= \sum_{x\in\omega_{\Delta_{l+1}\setminus\Delta_l}} \int_{L^R_{\Delta}(\omega)} \phi(\Vert x-y\Vert) \,{\rm d} y \\[2pt]&\quad + \sum_{x\in\omega_{\Delta}}\int_{L^R(\omega_{\Delta_{l+1}})\setminus L^R(\omega_{\Delta_l})} \phi(\Vert x -y\Vert)\,{\rm d} y.\end{align*}

If we choose $\Delta_l = B(z,\rho+2R+l)$ , for $x\in\omega_{\Delta_{l+1}\setminus\Delta_l}$ and $y \in L^R_{\Delta}(\omega)$ or for $x\in\omega_{\Delta}$ and $y\in L^R(\omega_{\Delta_{l+1}})\setminus L^R(\omega_{\Delta_l})$ , then we have $\Vert x - y \Vert > A + l$ . Then, we have

\begin{equation*} \vert H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) \vert \leq \left(\vert \omega_{\Delta_{l+1}\setminus\Delta_{l}} \vert \lambda^d\big(L^R_{\Delta}(\omega)\big) + \vert\omega_{\Delta}\vert \lambda^d\big(L^R(\omega_{\Delta_{l+1}\setminus\Delta_l})\big) \right) \psi(A+l).\end{equation*}

With the union bound we have

\begin{equation*} \lambda^d\big(L^R_\Delta(\omega)\big) \leq \vert \omega_\Delta \vert c_dR^d \text{ and }\lambda^d\big(L^R(\omega_{\Delta_{l+1}\setminus\Delta_l})\big) \leq \vert \omega_{\Delta_{l+1}\setminus\Delta_l}\vert c_dR^d.\end{equation*}

If we introduce the function $G^l_\Delta(\omega_{\Delta^{\rm c}}) = 2 \vert \omega_{\Delta_{l+1}\setminus\Delta_l} \vert c_d R^d \psi(A+l)$ , we obtain $\vert H^{l+1}_{\Delta}(\omega) - H^l_{\Delta}(\omega) \vert \leq \vert \omega_{\Delta} \vert G^l_{\Delta}(\omega_{\Delta^{\rm c}})$ and, for all P with bounded intensity,

\begin{align*} \mathbb{E}_P\left[G^l_{\Delta}(\omega_{\Delta^{\rm c}})\right] &\leq \xi(P)\lambda^d(\Delta_{l+1}\setminus\Delta_l) \psi(A+l) \\[2pt] & \leq \xi(P) d c_d(\rho+A+l+1)^{d-1} \psi(A+l).\end{align*}

Since $\int_A^{+\infty}\psi(r)r^{d-1}\,{\rm d} r<+\infty$ and $\psi$ is decreasing, this implies that

\begin{align*} \sum_{l\geq0}(\rho+A+l+1)^{d-1} \psi(A+l) < +\infty.\end{align*}

Hence, the energy H satisfies assumption [Intensity Regular].

4.1. Construction of an infinite volume measure

The first step of the proof is to build an accumulation point of a sequence of finite-volume Gibbs measures. Using entropy bounds and the stability of the energy, we prove the existence of such an accumulation point for the local convergence topology. This strategy and its tools have been used several times in the literature [Reference Dereudre3, Reference Dereudre4, Reference Dereudre, Drouilhet and Georgii5, Reference Dereudre and Houdebert6], and we recall here only the main ideas.

For n a positive integer, we denote by $\Lambda_n$ the set $ ]-n,n]^d$ . We consider the sequence of Gibbs measures in finite volume given by

\begin{equation*}P_n({\rm d}\omega) = P_{\Lambda_n}({\rm d}\omega) = \frac{1}{Z_n} {\rm e}^{- H(\omega)} \pi_{\Lambda_n}({\rm d}\omega),\end{equation*}

with the normalization constant $Z_n = \int {\rm e}^{- H(\omega)} \pi_{\Lambda_n}({\rm d}\omega)$ . Since our tension tool will be defined for stationary measures, we need to modify $(P_n)_{n\geq1}$ . We defined the periodized version $P_n^{\rm per}$ by the probability measure $\underset{u\in\mathbb{Z}^d}{\bigotimes} P_n \circ \tau^{-1}_{2nu}$ , and the stationarized version by

\begin{equation*}P^{\textrm{sta}}_n = \frac{1}{\lambda^d(\Lambda_n)}\int_{\Lambda_n} P_n^{\textrm{per}} \circ \tau^{-1}_{u} \,{\rm d} u.\end{equation*}

Definition 6. A function f is said to be local if there exists a bounded set $\Delta$ such that f is $\mathcal{F}_{\Delta}$ measurable (i.e. for all configurations $\omega$ in $\Omega$ , we have $f(\omega) = f(\omega_{\Delta})$ ). A sequence of measures $(\mu_n)$ converges to $\mu$ for the local convergence topology if, for all bounded local functions f,

\begin{equation*}\int f \,{\rm d}\mu_n \underset{n\rightarrow +\infty}{\rightarrow} \int f \,{\rm d}\mu.\end{equation*}

Given two probabilities measures $\mu$ and $\nu$ on $\Omega$ , we recall that the relative entropy of $\mu$ with respect to $\nu$ on $\Lambda$ is defined as

\begin{equation*}I_{\Lambda}(\mu \mid \nu) = \left\{ \begin{array}{l@{\quad}l} \int \log f \,{\rm d}\mu_{\Lambda} & \mbox{if } \mu_{\Lambda} \ll \nu_{\Lambda} \mbox{ and } f = \frac{{\rm d}\mu_{\Lambda}}{{\rm d}\nu_{\Lambda}}, \\[3pt] +\infty & \mbox{otherwise.} \end{array}\right.\end{equation*}

Definition 7. Let $\mu$ be a stationary probability measure with finite intensity on $\Omega$ . For $z > 0$ the specific entropy of $\mu$ with respect to $\pi^{z}$ is defined by

(3) \begin{equation}I_{z}(\mu \mid \pi^{z}) = \lim_{n\rightarrow +\infty} \frac{I_{\Lambda_n}(\mu \mid \pi^{z})}{\lambda^d(\Lambda_n)}=\sup_{\substack{\Lambda \subset \mathbb{R}^d \\ 0 < \lambda^d(\Lambda) < +\infty}} \frac{I_\Lambda(\mu\mid\pi^z)}{\lambda^d(\Lambda)}.\end{equation}

For details we refer to [Reference Georgii7, Chapter 15]. The next result, stated in [Reference Georgii and Zessin9], is our tension tool.

Proposition 1. For any $z > 0$ and $c > 0$ , the set of probability measures

\begin{equation*}\{ \mu\ {stationary\ with\ finite\ intensity,}\ I_{z}(\mu) \leq c\}\end{equation*}

is compact and sequentially compact for the local convergence topology.

In order to apply this proposition in our case, we need to compute the specific entropy of the probability measure $P^{\textrm{sta}}_n$ . Using the affine property of the specific entropy, it is well known that

\begin{equation*}I_{z}(P^{\textrm{sta}}_n)= \frac{1}{\lambda^d(\Lambda_n)} I_{\Lambda_n}(P_n \mid \pi^{z}).\end{equation*}

What remains is to compute the relative entropy of the Gibbs measure $P_n$ with respect to the Poisson point process $\pi^{z}$ on $\Lambda_n$ :

\begin{align*}I_{\Lambda_n}(P_n \mid \pi^{z})& = \int \log\!\left(\frac{{\rm d} P_n}{{\rm d}\pi^{z}}\right) {\rm d} P_n \\& = \int \left[\log\!\left(\frac{{\rm d} P_n}{{\rm d}\pi}\right) + \log\!\left(\frac{{\rm d}\pi}{{\rm d}\pi^{z}}\right)\right] {\rm d} P_n\\& = \int \left[-\log(Z_n) - H(\omega) + \log\!\left({\rm e}^{(z - 1)\lambda^d(\Lambda_n)}\left(\frac{1}{z}\right)^{\vert \omega \vert}\right)\right] P_n({\rm d}\omega) \\&= (z - 1) \lambda^d(\Lambda_n) - \log(Z_n) + \int\left[ - H(\omega) - \log(z) \vert \omega \vert \right] P_n({\rm d}\omega).\end{align*}

The normalization constant can easily be bounded from below:

\begin{equation*}Z_n = \int {\rm e}^{- H} \,{\rm d}\pi \geq {\rm e}^{- H(\emptyset)} {\rm e}^{-\lambda^d(\Lambda_n)}.\end{equation*}

Then, using the stability of H, we have

\begin{equation*}I_{\Lambda_n}(P_n \mid \pi^{z}) \leq z \lambda^d(\Lambda_n) + H(\emptyset) + \int (B - \log(z) )\vert \omega \vert P_n({\rm d}\omega).\end{equation*}

If we choose $z>0$ such that $B - \log(z) \leq 0$ , we have $I_{z}(P^{\textrm{sta}}_n) \leq z + H(\emptyset)$ for all $n\geq 1$ . According to Proposition 1 we can exhibit a subsequence of $(P_n^{\textrm{sta}})_{n\geq 1}$ which converges to a stationary measure P with finite intensity. To simplify the notation we can suppose that we have changed the indexation of the sequence $(\Lambda_n)_{n\geq 1}$ such that $(P^{\textrm{sta}}_n)_{n\geq 1}$ converges locally to P.

We can prove that P is also an accumulation point of the sequence

\begin{equation*}\bar{P}_n = \frac{1}{\lambda^d(\Lambda_n)} \int_{\Lambda_n} P_n \circ \tau^{-1}_{u} \,{\rm d} u.\end{equation*}

See [Reference Dereudre3, Lemma 3.5] for details.

We need to verify that P has locally finite energy. If $\Delta$ is a bounded subset of $\mathbb{R}^d$ , the event $\{H(\omega_\Delta) < +\infty\}$ is a local event and

\begin{equation*}P(H(\omega_\Delta) <+\infty) = \lim_{n\rightarrow +\infty} \bar{P}_n(H(\omega_\Delta)<+\infty) = 1,\end{equation*}

because by heredity we have $\bar{P}_n(H(\omega_\Delta) =+\infty) = 0$ . We deduce, again using heredity, that

\begin{equation*}P\!\left(H(\omega_\Delta) < +\infty,\, \text{for all } \Delta \subset \mathbb{R}^d, \text{ bounded}\right) = P\!\left(H(\omega_{\Lambda_n})) < +\infty,\, \text{for all } n \geq 1\right) = 1.\end{equation*}

We finish this section by giving the crucial property of uniform control of intensities for the sequence $(\bar P_n)$ .

Proof. We use the entropic inequality $\int g \,{\rm d}\mu \leq I(\mu\mid\nu) + \log\!\left(\int {\rm e}^g \,{\rm d}\nu \right)$ to obtain

\begin{equation*}\mathbb{E}_{P^{\textrm{sta}}_n}[\vert \omega_{\Lambda_n} \vert] \leq I_{\Lambda_n}(P^{\textrm{sta}}_n \mid \pi^{z}) + \log \left(E_{\pi^{z}}\big[{\rm e}^{\vert \omega_{\Lambda_n} \vert}\big]\right).\end{equation*}

Using the expression of the specific entropy as a supremum (3), we have

\begin{equation*}I_{\Lambda_n}\big(P^{\textrm{sta}}_n \mid \pi^{z}\big) \leq I_{z}\big(P^{\textrm{sta}}_n\big) \lambda^d(\Lambda_n) \leq (z + H(\emptyset))\lambda^d(\Lambda_n).\end{equation*}

Under $\pi^{z}$ , the random variable $\vert \omega_{\Lambda_n} \vert$ follows a Poisson law of parameter $z \lambda^d(\Lambda_n)$ , so

\begin{equation*}\mathbb{E}_{\pi^{z}}\big[{\rm e}^{\vert \omega_{\Lambda_n}\vert} \big] = {\rm e}^{-z \lambda^d(\Lambda_n)}\sum_{p=0}^{\infty} \frac{\big(z\lambda^d(\Lambda_n)\big)^p}{p!}{\rm e}^p = \exp(z \lambda^d(\Lambda_n)({\rm e}-1)).\end{equation*}

Then, we obtain

\begin{equation*}i\big(P^{\textrm{sta}}_n\big) \lambda^d(\Lambda_n) = \mathbb{E}_{P^{\textrm{sta}}_n}[\vert \omega_{\Lambda_n} \vert] \leq (z {\rm e} + H(\emptyset))\lambda^d(\Lambda_n),\end{equation*}

and since $\xi(\bar{P}_n) \leq i(P^{\textrm{sta}}_n)$ , we deduce the lemma.

4.2. Some remarks on the local energy

Lemma 2. If P is a point process with bounded intensity, then for P-almost all $\omega$ the following equivalence holds for all $l\geq 0$ :

\begin{equation*} H_\Delta(\omega) = + \infty \iff H^l_\Delta(\omega) = +\infty.\end{equation*}

Proof. Let $\omega$ be a configuration. With our convention $\infty - \infty = 0$ and the intensity regular assumption, for $l\geq 1$ , having $H^l_\Delta(\omega) = +\infty$ implies that $H^{l-1}_\Delta(\omega) = +\infty$ and $H^{l+1}_\Delta(\omega) = +\infty$ . Then, by induction we deduce that, for all $l\geq 1$ , we have $H^l_\Delta(\omega) = +\infty$ if and only if $H^0_\Delta(\omega) = +\infty$ .

The local energy is defined by

\begin{equation*} H_\Delta(\omega) = H_\Delta^0(\omega) + \sum_{l=0}^{+\infty} \left[H^{l+1}_\Delta(\omega) - H^l_\Delta(\omega)\right],\end{equation*}

and since for a point process P with bounded intensity the second part of the right-hand side is finite, we deduce that, for P-almost all $\omega$ , $H_\Delta(\omega) = +\infty$ if and only if $H_\Delta^0(\omega) = +\infty$ .

Lemma 2 ensures that the event $\{H_\Delta(\omega) = +\infty\}$ is local. In particular, the hard-core part of the energy has to be finite range.

For a finite configuration $\omega\in\Omega_{\rm f}$ , the sum in the definition of the local energy $H_\Delta(\omega)$ is a finite sum, and then Lemma 2 also holds. Hence, we can write

\begin{equation*}H_\Delta(\omega) = H^l_\Delta(\omega) = H(\omega_{\Delta_l}) - H(\omega_{\Delta_l^c}) = H(\omega) - H(\omega_{\Delta^{\rm c}})\end{equation*}

for some $l\geq0$ (depending on $\omega$ ). Moreover, thanks to the heredity of the energy and our convention $\infty - \infty = 0$ , we have, for all $\omega\in\Omega_{\rm f}$ ,

\begin{equation*}H(\omega) = H_\Delta(\omega) + H(\omega_{\Delta^{\rm c}}).\end{equation*}

This equality is useful for proving the DLR equations for finite-volume Gibbs measures.

4.3. The DLR equation

We prove in this section that the accumulation point P satisfies the DLR equations stated in Definition 5. By a standard monotone class argument we can replace the class of bounded measurable functions by the class of bounded local functions. Let f be a bounded local function and $\Delta$ be a bounded measurable subset of $\mathbb{R}^d$ ; we have to show that $\int f \,{\rm d} P = \int f_{\Delta} \,{\rm d} P$ with

\begin{equation*}f_{\Delta}(\omega) = \int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}) \frac{1}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})} {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}).\end{equation*}

We fix $\varepsilon > 0$ .

Step 1 There exists $K' > 0$ such that, for all $k \geq K'$ , we have $P(\vert\omega_{\Delta}\vert > k) \leq \varepsilon$ ; this implies that, for all $k \geq K'$ ,

(4) \begin{equation}\left| \int f_{\Delta}(\omega)P({\rm d}\omega) - \int f_{\Delta}(\omega){\textbf{1}}_{\vert\omega_{\Delta}\vert\leq k} P({\rm d}\omega) \right| \leq \Vert f \Vert_{\infty}^{}\varepsilon.\end{equation}

The introduction of this indicator function will be useful in Step 4.

Step 2 We approximate $f_{\Delta}$ by $f_{\Delta, k}^{l}$ , which corresponds to the approximation of the local energy $H_{\Delta}$ by $H_{\Delta}^l$ and a restriction to configurations having less than k points in $\Delta$ , which means

\begin{equation*}f_{\Delta, k}^{l}(\omega) = \frac{1}{Z^{l}_{\Delta, k}(\omega_{\Delta^{\rm c}})}\int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}){\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}),\end{equation*}

with the normalization constant $Z^{l}_{\Delta, k}(\omega_{\Delta^{\rm c}}) = \int {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta})$ . We prove that we can find $K \geq K'$ and l (depending on K) such that

(5) \begin{equation}\left|\int f_{\Delta}(\omega){\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} P({\rm d}\omega) - \int f_{\Delta, K}^{l}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} P({\rm d}\omega) \right| \leq 6\Vert f \Vert_{\infty}^{}\varepsilon.\end{equation}

We introduce an event $\mathcal{A}_\varepsilon$ such that $P(\mathcal{A}_\varepsilon) \leq \varepsilon$ . Its precise definition is given later. We have

(6) \begin{align} & \left|\int f_{\Delta}(\omega){\textbf{1}}_{\vert\omega_{\Delta}\vert \leq k}^{} P({\rm d}\omega) - \int f_{\Delta, k}^{l}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} P({\rm d}\omega) \right| \\ &\quad \leq \int_{\mathcal{A}^c_\varepsilon} \vert f_{\Delta}(\omega) - f_{\Delta, k}^{l}(\omega) \vert {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq k}^{} P({\rm d}\omega) + 2\Vert f \Vert_{\infty}^{}\varepsilon. \nonumber\end{align}

We must estimate the approximation error

\begin{align*}& f_{\Delta}(\omega) - f_{\Delta, k}^{l}(\omega) \\[3pt] & = \frac{1}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}) \left({\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} - {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{}\right) \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \\[3pt] & \quad + \Bigg(\frac{1}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}-\frac{1}{Z^{l}_{\Delta, k}(\omega_{\Delta^{\rm c}})}\Bigg)\int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}) {\rm e}^{- H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}).\end{align*}

Since the difference between the normalization constants is

\begin{equation*}Z^{l}_{\Delta, k}(\omega_{\Delta^{\rm c}})-Z_{\Delta}(\omega^{}_{\Delta^{\rm c}}) = \int \left({\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}- {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}\right) \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}),\end{equation*}

we obtain the upper bound

(7) \begin{align}\vert f_{\Delta}(\omega) - f_{\Delta, k}^{l}(\omega)\vert \leq {} & \frac{2\Vert f \Vert_{\infty}^{}}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int \left\vert {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} - {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} \right\vert\pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \nonumber\\[3pt] = {} & \frac{2\Vert f \Vert_{\infty}^{}}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int \left\vert {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} - {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} \right\vert {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \nonumber\\[3pt] & + \frac{2\Vert f \Vert_{\infty}^{}}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert > k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}).\end{align}

Using Lemma 2, the inequality $\vert {\rm e}^b- {\rm e}^a \vert \leq \vert b -a \vert {\rm e}^{\vert b - a\vert + a} $ and the approximation (1), we obtain the upper bound

(8) \begin{equation}\left\vert {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} - {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} \right\vert {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K}^{}\leq K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}.\end{equation}

By combining (7) and (8), we have

(9) \begin{align} &\vert f_{\Delta}(\omega) - f_{\Delta, k}^{l}(\omega) \vert \leq 2 \Vert f \Vert_{\infty}^{} K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} \\ & \quad + \frac{2\Vert f \Vert_{\infty}^{}}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert > k}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}).\nonumber\end{align}

Now we need to control the two terms of the right-hand side of (9). For the second one, by the dominated convergence theorem, we can find $K \geq K'$ such that

\begin{equation*}\int\frac{1}{Z_{\Delta}(\omega^{}_{\Delta^{\rm c}})}\int {\rm e}^{- H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert > K}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta})P({\rm d}\omega) \leq \varepsilon.\end{equation*}

Once K is chosen, we can control the first term. If $W\,{:}\,\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is the inverse function of $x \mapsto x{\rm e}^x$ (W is the so-called Lambert function), using Markov’s inequality we have

\begin{equation*} P\!\left( K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} > \varepsilon\right) = P(K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) > W(\varepsilon)) \leq \frac{K \mathbb{E}_P\big[C^l_{\Delta}(\omega_{\Delta^{\rm c}})\big]}{W(\varepsilon)}.\end{equation*}

As $\mathbb{E}^P[C^l_{\Delta}(\omega_{\Delta^{\rm c}})]$ goes to zero when l goes to infinity, we can choose l (depending on K) such that

\begin{equation*}P\!\left( K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} > \varepsilon\right) \leq \varepsilon.\end{equation*}

Introducing the event $\mathcal{A}_\varepsilon = \{K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} > \varepsilon \}$ in (6), and with our choice of K and l, we finally have the approximation (5).

According to Lemma 1, the point processes $(\bar{P}_n)_{n\geq1}$ have uniformly bounded intensities and, as mentionned in Remark 1, the expectation $\mathbb{E}_{\bar{P}_n}[C^l_{\Delta}(\omega_{\Delta^{\rm c}})]$ is bounded from above uniformly in $n\geq1$ . Hence, l could be chosen such that, for all $n\ge 1$ ,

(10) \begin{equation}\bar{P}_n\left(K C^l_{\Delta}(\omega_{\Delta^{\rm c}}) {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} > \varepsilon\right) \leq \varepsilon . \end{equation}

This will be useful later, in Step 4.

Step 3 For n large enough (depending on K and l), we have

(11) \begin{equation}\left|\int f_{\Delta, K}^{l}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} P({\rm d}\omega) - \int f_{\Delta, K}^{l}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} \bar{P}_n({\rm d}\omega) \right| \leq \Vert f \Vert_{\infty}^{}\varepsilon.\end{equation}

This is simply a consequence of the local convergence of the sequence $(\bar{P}_n)_{n\geq 1}$ to P.

Step 4 For all $n \geq 1$ we show the approximation

(12) \begin{equation}\left| \int f_{\Delta, K}^{l}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} \bar{P}_n({\rm d}\omega) - \int f_{\Delta, K}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} \bar{P}_n({\rm d}\omega) \right| \leq 4 \Vert f \Vert_{\infty}^{} \varepsilon,\end{equation}

where

\begin{equation*}f_{\Delta, K}(\omega) = \frac{1}{Z_{\Delta, K}(\omega_{\Delta^{\rm c}})}\int f(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}}){\rm e}^{-H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}),\end{equation*}

with the normalization constant $Z_{\Delta, K}(\omega_{\Delta^{\rm c}}) = \int {\rm e}^{-H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K}^{} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta})$ .

Similarly to the upper bounds (7) and (8), we obtain

\begin{align*}\vert f_{\Delta, K}^{l}(\omega) - f_{\Delta, K}(\omega) \vert \leq {} & \frac{2\Vert f \Vert_{\infty}^{}}{Z_{\Delta, K}(\omega_{\Delta^{\rm c}})}\int \big\vert {\rm e}^{-H_{\Delta}(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})} - {\rm e}^{-H_{\Delta}^l(\omega^{\prime}_{\Delta}\omega_{\Delta^{\rm c}})}\big\vert {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \\[3pt] \leq {} & 2\Vert f \Vert_{\infty}^{} {\rm e}^{ K C^l_{\Delta}(\omega_{\Delta^{\rm c}})} K C^l_{\Delta}(\omega_{\Delta^{\rm c}}).\end{align*}

From our previous choice of K and l in estimate (10), and using a restriction to a specific event as in (6), we obtain the approximation (12).

Step 5 We use the DLR equations for finite-volume Gibbs processes to prove that

(13) \begin{equation}\left| \int f_{\Delta, K}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n({\rm d}\omega) - \int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n({\rm d}\omega) \right|\leq 2 \Vert f \Vert_{\infty}^{} \varepsilon.\end{equation}

Let us introduce $\Lambda_n^* = \{ u \in \Lambda_n\,{:}\,\tau^{-1}_{u}(\Delta) \subset \Lambda_n \}$ . Note that if $\Delta \subset \Lambda_k$ and $n \geq k$ then $\Lambda_{n-k} \subset \Lambda_n^*$ and $(n-k)^d / n^d \leq \lambda^d(\Lambda_n^*) / \lambda^d(\Lambda_n) \leq 1$ . We choose n large enough such that $\lambda^d(\Lambda_n^*) / \lambda^d(\Lambda_n) \geq 1 - \varepsilon$ , and if we denote

\begin{equation*}\bar{P}_n^* = \frac{1}{\lambda^d(\Lambda_n)}\int_{\Lambda_n^*} P_n \circ \tau^{-1}_{u},\end{equation*}

we have the approximation

\begin{equation*}\left| \int f_{\Delta, K}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} \bar{P}_n({\rm d}\omega) - \int f_{\Delta, K}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K} \bar{P}_n^*({\rm d}\omega)\right| \leq \Vert f \Vert_{\infty}^{} \varepsilon.\end{equation*}

Let us detail the term

\begin{align*}& \int f_{\Delta, K}(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n^*({\rm d}\omega) \\[3pt]&\quad = \frac{1}{\lambda^d(\Lambda_n)} \int_{\Lambda_n^*} \int f_{\Delta, K}(\tau_u(\omega)){\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} P_n({\rm d}\omega) \,{\rm d} u \\[3pt] &\quad = \frac{1}{\lambda^d(\Lambda_n)} \int_{\Lambda_n^*} \iint f(\omega^{\prime}_{\Delta}\tau_u(\omega)_{\Delta^{\rm c}}) \frac{1}{Z_{\Delta, K}(\tau_u(\omega)_{\Delta^{\rm c}}^{})} {\rm e}^{-H_{\Delta}(\omega^{\prime}_{\Delta} \tau_u(\omega)_{\Delta^{\rm c}})} {\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K}^{}\pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \\[3pt] &\qquad {\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} P_n({\rm d}\omega) \,{\rm d} u. &\end{align*}

For $u\in\Lambda_n^*$ , using the fact that $\tau_u(\omega)_{\Delta^{\rm c}} = \tau_u\big(\omega_{\tau^{-1}_{u}(\Delta)^c}\big) $ , that $\pi_{\Delta}$ has the same law as $\pi_{\tau^{-1}_{u}(\Delta)} \circ \tau^{-1}_{u}$ , and that $H_{\Delta}(\tau_u(\omega)) =H_{\tau^{-1}_u(\Delta)}(\omega)$ , we have

\begin{align*} Z_{\Delta, K}(\tau_u(\omega)_{\Delta^{\rm c}}^{})& = \int {\rm e}^{-H_{\Delta}\big(\omega^{\prime}_{\Delta}\tau_u\big(\omega_{\tau^{-1}_{u}(\Delta)^c}\big)\big)}{\textbf{1}}_{\vert\omega^{\prime}_{\Delta}\vert \leq K} \pi_{\Delta}({\rm d}\omega^{\prime}_{\Delta}) \\[3pt] & = \int {\rm e}^{-H_{\Delta}\big(\tau_u\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)\big)}{\textbf{1}}_{\big|\omega'_{\tau^{-1}_{u}(\Delta)}\big| \leq K} \pi_{\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega'_{\tau^{-1}_{u}(\Delta)}\big)\\[3pt] & = \int {\rm e}^{-H_{\tau^{-1}_{u}(\Delta)}\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)}{\textbf{1}}_{\big|\omega'_{\tau^{-1}_{u}(\Delta)}\big| \leq K} \pi_{\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega'_{\tau^{-1}_{u}(\Delta)}\big)\\[3pt] & = Z_{\tau^{-1}_{u}(\Delta), K}\big(\omega_{\tau^{-1}_{u}(\Delta)^c}\big).\end{align*}

Then, by a similar calculation, we find

\begin{align*} &\int f_{\Delta, K}(\tau_u(\omega)){\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} P_n({\rm d}\omega) \\[3pt] &\quad =\iint f\big(\tau_u\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)\big) \frac{1}{Z_{\tau^{-1}_{u}(\Delta), K}\big(\omega_{\tau^{-1}_{u}(\Delta)^c}\big)} {\rm e}^{- H_{\tau^{-1}_{u}(\Delta)}\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)} \\[3pt] &\qquad {\textbf{1}}_{\vert\omega'_{\tau^{-1}_{u}(\Delta)}\vert \leq K}^{} \pi_{\tau^{-1}_{u}(\Delta)}({\rm d}\omega'_{\tau^{-1}_{u}(\Delta)}) {\textbf{1}}_{\vert\omega_{\tau^{-1}_{u}(\Delta)}\vert \leq K}^{} P_n({\rm d}\omega).\end{align*}

But we can write the measure in finite volume as

\begin{align*}P_n(d\omega) &= \frac{1}{Z_n} {\rm e}^{- H_{\tau^{-1}_{u}(\Delta)}\big(\omega_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)}{\rm e}^{- H\big(\omega_{\tau^{-1}_{u}(\Delta)^c}\big)} \\[3pt] &\quad \pi^z_{\Lambda_n\setminus\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega_{\tau^{-1}_{u}(\Delta)^c}\big) \pi_{\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega_{\tau^{-1}_{u}(\Delta)}\big),\end{align*}

and integration with respect to the measure $\pi_{\tau^{-1}_{u}(\Delta)}$ will give the normalization constant (thanks to the indicator function introduced in Step 1). After simplification we have for the translated $P_n \circ \tau^{-1}_{u}$ with $u\in\Lambda^*_n$ a finite-volume DLR equation:

\begin{align*}&\int f_{\Delta, K}(\tau_u(\omega)){\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} P_n({\rm d}\omega) \\[3pt] &\quad = \frac{1}{Z_n} \iint f\big(\tau_u\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)\big) {\rm e}^{- H\big(\omega'_{\tau^{-1}_{u}(\Delta)}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)} {\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} \\[3pt] & \qquad \pi^z_{\Lambda_n\setminus\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega_{\tau^{-1}_{u}(\Delta)^c}\big)\pi_{\tau^{-1}_{u}(\Delta)}\big({\rm d}\omega'_{\tau^{-1}_{u}(\Delta)}\big) \\[3pt] &\quad = \int f(\tau_u(\omega)){\textbf{1}}_{\vert\tau_u(\omega)_{\Delta}\vert \leq K}^{} P_n({\rm d}\omega).\end{align*}

This DLR-type equation is then verified for the mixture $\bar{P}_n^*$ :

\begin{equation*}\int f_{\Delta, K}(\omega){\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n^*({\rm d}\omega) = \int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n^*({\rm d}\omega).\end{equation*}

Since we have the approximation

\begin{equation*}\left| \int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n^*({\rm d}\omega) - \int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n({\rm d}\omega) \right| \leq \Vert f \Vert_{\infty}^{} \varepsilon,\end{equation*}

we finally obtain (13).

Step 6 We show the last approximation,

(14) \begin{equation}\left\vert\int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n({\rm d}\omega) - \int f(\omega) P({\rm d}\omega) \right\vert \leq 2\Vert f \Vert_{\infty}^{}\varepsilon.\end{equation}

Using the local convergence of $(P_n)_{n\geq 1}$ to P again, we have, for n large enough,

\begin{equation*}\left\vert\int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} \bar{P}_n({\rm d}\omega) - \int f(\omega) {\textbf{1}}_{\vert\omega_{\Delta}\vert \leq K}^{} P({\rm d}\omega) \right\vert \leq \Vert f \Vert_{\infty}^{}\varepsilon.\end{equation*}

With our choice of K we have $P(\vert\omega_{\Delta}\vert > K) \leq \varepsilon$ and we obtain (14).