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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, 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 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 $\Omega$.

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    $\begingroup$ The free b.c. measure on $\Omega$ is obtained by 1) defining the product measure in infinite volume, then 2) multiplying this infinite product measure (spins seen as independent Rademacher random variables) by the Radon-Nikodym weight $e^{-H_{\Lambda}}/Z_{\lambda}$. Intuitively this means the eta's outside the finite volume are random (independent $\pm 1$ spins) instead of deterministic as in the description of $\mu^{\eta}$. $\endgroup$ Aug 11 '20 at 18:53
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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.

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  • $\begingroup$ Amazing answer! Always good to talk to an expert! Just to clarify, the construction of limit states of $\mu_{\Lambda,\beta,h}^{\emptyset}$ is not precisely the same as for $\eta$ boundary conditions, since not all $\mu_{\Lambda,\beta,h}^{\emptyset}$ is defined to be in the same measurable space, right? One uses the convergence of real numbers $\langle f\rangle_{\Lambda_{n},\beta,h}^{\emptyset}$ to a real number $\langle f \rangle_{\beta,h}^{\emptyset}$ for each local $f$ and then one uses the Riesz-Markov-Kakutani representation theorem. Is that right? $\endgroup$
    – MathMath
    Aug 11 '20 at 16:47
  • $\begingroup$ Yes, exactly. Of course, nothing prevents you from extending a configuration in $\Omega_\Lambda$ to a configuration in $\Omega$ by fixing the value of all spins outside $\Lambda$ to be $+$, for example. Since the measure $\mu_{\Lambda,\beta,h}^\varnothing$ does not feel at all what's happening outside $\Lambda$, this does not change anything, but it makes $f$ (and the expectations) to be always well defined. But I find it simpler to just consider a sequence of large enough boxes ;) . $\endgroup$ Aug 11 '20 at 16:53
  • $\begingroup$ great! Perfect! I got it now! Thank you so much! $\endgroup$
    – MathMath
    Aug 11 '20 at 16:56

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