This is actually a more elaborated version of a previous question of mine, which is now deleted. First, some quick notations:


**(1)** $\Omega\_{0} := \{-1,1\}$ and $\mathcal{F}\_{0} := 2^{\Omega\_{0}}$ are, respectivelly, the single particle configuration space and its associated $\sigma$-algebra.


**(2)** If $\Lambda \subset \mathbb{Z}^{d}$ is finite, $\Omega\_{\Lambda} := \{-1,1\}^{\Lambda}$ and $\mathcal{F}\_{\Lambda} := \bigotimes\_{x\in \Lambda}\mathcal{F}\_{0}$ is its associated product $\sigma$-algebra.


**(3)** $\Omega := \{-1,1\}^{\mathbb{Z}^{d}}$, $\mathcal{F} := \bigotimes\_{x\in \mathbb{Z}^{d}}\mathcal{F}\_{0}$.


In what follows, I'm following [Friedli and Velenik's book](http://www.unige.ch/math/folks/velenik/smbook/Ising_Model.pdf), chapter 3. For finite-volume systems, we can define Gibbs distributions with free boundary conditions (according to Definition 3.1 of the mentioned reference) by:
\begin{eqnarray}
\mu\_{\Lambda,\beta,h}^{\emptyset}(\{\omega\}):= \frac{1}{Z\_{\Lambda,\beta, h}^{\emptyset}}e^{-\beta H\_{\Lambda,\beta,h}^{\emptyset}(\omega)} \tag{1}\label{1}
\end{eqnarray}
this is a discrete measure on $\Omega\_{\Lambda}$. Also, if we fix $\eta \in \Omega$, we can define (according to Definition 3.3) Gibbs states with $\eta$-boundary conditions:
\begin{eqnarray}
\mu\_{\Lambda,\beta,h}^{\eta}(\{\omega\}):= \frac{1}{Z\_{\Lambda,\beta, h}^{\eta}}e^{-\beta H\_{\Lambda,\beta,h}^{\eta}(\omega)} \tag{2}\label{2}
\end{eqnarray}
This, on the other hand, is a discrete measure on $\Omega\_{\Lambda}^{\eta}:= \{\omega \in \Omega: \hspace{0.1cm} \mbox{$\omega\_{x} = \eta\_{x}$ for all $x \in \Lambda^{c}$}\}$ (which is equipped with its discrete $\sigma$-algebra).


The main ideia of the theory is to study thermodynamic limits, both for thermodynamic quantities and Gibbs states. In the case of Gibbs states, thermodynamic limits means weak convergence of finite-volume Gibbs states (at least in the present context, where $\Omega\_{0}=\{-1,1\}$. This is, in fact, the main reason for which $\mu\_{\Lambda,\beta,h}^{\eta}$ is conveniently defined on $\Omega\_{\Lambda}^{\eta}$ instead of $\Omega\_{\Lambda}$. Once we'd like to study Gibbs states on $\Omega$ using weak-convergence of finite volume Gibbs states, we need to extend our finite volume Gibbs states to equivalent notions that live in 'the whole space' $\Omega$. It is easy to do that with the measure (\ref{2}), since we can define (with abuse of notation):
\begin{eqnarray}
\mu\_{\Lambda, \beta,h}^{\eta}(\{\omega\}) = \begin{cases}
\displaystyle \frac{1}{Z\_{\Lambda,\beta, h}^{\eta}}e^{-\beta H\_{\Lambda, h}^{\eta}(\Pi\_{\Lambda,\eta}\omega)} \quad \mbox{if $\omega\_{x} = \eta\_{x}$ for all $x\in \Lambda^{c}$} \\
\displaystyle 0 \quad \mbox{otherwise}
\end{cases}
\tag{3}\label{3}
\end{eqnarray}
where $\Pi\_{\Lambda,\eta}$ is the canonical projection $\Omega \hookrightarrow \Omega\_{\Lambda}^{\eta}$. Note that this simple procedure does not work for extending $\mu\_{\Lambda,\beta,h}^{\emptyset}$, since this would require to define $\eta$ as zero outside $\Lambda$, which is not consistent with our definition of $\Omega\_{0}$.


Now, in this context, [R. Ellis](https://www.springer.com/gp/book/9783540290599) defines (I think this is standard, tho) $\mathcal{G}\_{0}(\beta,h)$ as the set of all weak-limits of measures $\mu\_{\Lambda\_{n},\beta,h}^{\eta\_{n}}$, where $(\forall n)$ $\eta\_{n}\in \Omega$ and $\Lambda\_{n}\to \mathbb{Z}^{d}$ is an increasing sequence of finite subsets of $\mathbb{Z}^{d}$. Furthermore, we define:
\begin{eqnarray}
\mathcal{G}(\beta,h) := \overline{\mbox{conv}\mathcal{G}\_{0}(\beta,h)} \tag{4}\label{4}
\end{eqnarray}
where $\mbox{conv}\mathcal{G}\_{0}(\beta,h)$ stands for the convex hull of $\mathcal{G}\_{0}(\beta,h)$.


**Question(s):** First, I'd like to know how to extend $\mu\_{\Lambda,\beta,h}^{\emptyset}$ to 'the whole space' $\Omega$ as done in (\ref{3}), since we cannot take zero values on $\Omega\_{0}$. Also, I'd like to understand the role of these measures $\mu\_{\Lambda,\beta,h}^{\emptyset}$ on $\Omega$: there are some results on weak-convergence of these measures in the theory, which seems to indicate they play some important role after all, but they seem **not being considered** in the definition of $\mathcal{G}\_{0}(\beta,h)$, since we're only taking $\eta\_{n}\in \Omega$ as boundary conditions of the sequence of Gibbs states $\mu\_{\Lambda\_{n},\beta,h}^{\eta\_{n}}$. So, what am I missing here? Can we go on and study all (or at least almost all) relevant issues of the theory without considering infinite volume measures $\mu\_{\Lambda,\beta,h}^{\emptyset}$? I know these measures are relevant to study, say, correlation inequalities and so on, but these do not demand them to be defined in the whole $\Omega$.



What is the role of Gibbs states with free boundary conditions in the theory of Gibbs measure?
One way to construct the thermodynamic limit of the states $\mu\_{\Lambda,\beta,h}^\varnothing$ is to observe that, for any local function $f$ and any increasing sequence of sets $\Lambda\_n\uparrow\mathbb{Z}^d$, the support of $f$ will be included inside $\Lambda\_n$ for all large enough $n$. In particular, for any local function $f$, one can prove that the limit $\lim\_{n \to\infty}\mu\_{\Lambda\_n,\beta,h}^\varnothing(f)$ is well defined and independent of the sequence $(\Lambda\_n)$ (this is Exercise 3.16 in our book). Then, one shows that there is a single probability measure on $(\Omega,\mathcal{F})$ that satisfies
$$
\mu(f) = \lim\_{n\to\infty} \mu\_{\Lambda\_n,\beta,h}^\varnothing(f)
$$
for all local functions $f$ (this is Theorem 6.5 in our book). Finally, one proves that the measure $\mu$ indeed belongs to $\mathcal{G}(\beta,h)$ (this is Exercise 6.14 in our book); here $\mathcal{G}(\beta,h)$ is defined as the set of all probability measures satisfying the DLR equations (see the beginning of Chapter 6 in our book), which coincides with the definition you state (by Theorem 6.63 in the book).




---


Now, regarding the relevance of the free boundary condition. In my opinion, for the Ising model on $\mathbb{Z}^d$ (or other amenable graphs), the interest of this boundary condition is mostly technical (it's one of the few boundary conditions for which one can explicitly prove convergence, without resorting to compactness arguments). Moreover, the finite-volume measures enjoy nice properties that carry on to the limiting state and can occasionally be useful.


On nonamenable graphs, it can play a more important role. For instance, on trees it is known that, under some conditions, the state obtained by taking the thermodynamic limit using free boundary condition is extremal for a range of temperatures *below the critical temperature*.


Note that this boundary condition can be more interesting in other models. For instance, in the Potts model on $\mathbb{Z}^d$ with $q$ colors, when the phase transition is of first order (that is, when $d=2$ and $q\geq 5$, or when $d\geq 3$ and $q\geq 3$), then, *at the phase transition temperature*, the $q$ low-temperature pure states coexist with the (unique) high-temperature state. While the former can be selected by taking the thermodynamic limit using the corresponding monochromatic boundary condition, the latter can be selected using free boundary condition.


