A Phase Transition in a Quenched Amorphous Ferromagnet

Abstract

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over $\mathbb {R}^d$, $d\ge 2$. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the intensity of the underlying point process and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.

1 Introduction

1.1 Setup

In this paper, we study thermodynamic states of the following system. A countable collection of ‘particles’ is distributed over $\mathbb {R}^d$, $d\ge 2$, in such a way that every bounded $\Lambda \subset \mathbb {R}^d$ contains only finite number of them. Each ‘particle’ represents a cluster of negligible size of magnetically active physical particles. In our model, this amounts to assuming that a ‘particle’ is characterized by its location $x\in \mathbb {R}^d$ and spin $\sigma _x\in \mathbb {R}$. The locations of ‘particles’ constitute a locally finite set (configuration) $\gamma \subset \mathbb {R}^d$, and the spins take real values. We also assume that each $\sigma _x$ is characterized by one and the same symmetric a priori distribution $\chi$ on $\mathbb {R}$. The interaction is supposed to be pair-wise and attractive. For the ‘particles’ located at $x$ and $y$, the interaction energy is $\phi (|x-y|) \sigma _x \sigma _y$, with $\phi (r)\ge \phi _* >0$ for $r\in [0, r_*]$, and $\phi (r) =0$ for $r >r_*$. If $\gamma$ were a crystalline lattice, then the model would be a standard lattice system of ‘unbounded spins’. The study of Gibbs states of such spin systems goes back to the seminal paper [29], further continued in [6, 37, 38]. In [28], a similar model living on a more general discrete metric space was studied. The next step was made in [26] where the underlying set was a countable graph with globally unbounded vertex degrees. In that paper, a class of graphs was introduced in which vertices of large degree are sparse—a property formulated in [26] as a weighted summability of the vertex degrees. For such graphs, tempered Gibbs states of unbounded spin systems were constructed and studied. A natural continuation of those works would be to pass to random graphs, in which this kind of summability holds for almost all realizations. In the present paper we do this step by letting the underlying set $\gamma \subset \mathbb {R}^d$ be random, obeying the Poisson law with homogeneous density $\lambda >0$. The graph structure on $\gamma$ is then defined by the spin-spin interaction: $x, y\in \gamma$ are adjacent (neighbors) if $\phi (|x-y|) >0$. We call this model the amorphous ferromagnet, cf. [34, Sect. 11].

In view of the mentioned randomness, there can be two types of thermodynamic state of our model. In the first case, the randomness is taken into account already at the level of local states defined on the space of (marked) configurations $\hat{\gamma } = \{ (x, \sigma _x): x\in \gamma \}$. The global thermodynamic states constructed in this way are then annealed states; they describe the thermal equilibrium of the whole system. The second approach, which we follow in this paper, consists in constructing thermodynamic states of the spin system alone for fixed typical configurations $\gamma$. These are quenched states, cf. [8]. The global observables characterizing such states ought to be self-averaging – taking the same value for all typical (i.e., for almost all) configurations $\gamma$. Note that studying quenched states is a more difficult problem, as compared to that of annealed ones, since the present spatial irregularities do not allow for applying here most of the methods effective for regular systems. In what follows, we aim at proving:

• Existence, for almost all configurations $\gamma$, of thermodynamic states with properties suitable for physical applications.

• Measurability of thermodynamic states with respect to $\gamma$.

• Multiplicity of such states, for almost $\gamma$, for temperatures $T< T_*$, where the self-averaging parameter $T_*>0$ may depend on the model parameters $\lambda$, $\phi _*$, $r_*$, $d$.

There are only few publications on the mathematically rigorous theory of phase transitions in spin systems of general type living on non-crystalline (amorphous) substances, see [12, 20–22, 36] where annealed states were considered. The reason for this is presumably that the corresponding methods, e.g., infrared estimates, are essentially based on the translation invariance (and other symmetries) of the underlying crystals. At the same time, for Ising spins $\sigma _x = \pm 1$, there exist methods applicable to the corresponding models on graphs, see [23, 30]. For such models, see also [11], the main idea of proving the existence of phase transitions is to relate the appearance of multiple phases of the spin system to the Bernoulli bond percolation on the underlying graph. In our model, we deal with a random graph with vertex set $\gamma$ and the adjacency relation $x \sim y$ defined by the property $|x-y|\le r_*$. That is, the set of edges of the graph is $\varepsilon _\gamma =\{ \{x,y\} \subset \gamma : x \sim y\}$, and the graph itself—known as the Gilbert graph – is then the pair $(\gamma , \varepsilon _\gamma )$. It has various applications and is intensively studied, see, e.g., [5, 18, 31, 33]. The probability distribution of $(\gamma , \varepsilon _\gamma )$ is described in Sect. 2.1 below. If this graph has an infinite connected component, which is a random event with probability dependent on $\lambda$, under certain conditions one can observe the Bernoulli bond percolation with a nontrivial percolation threshold $q_*\in (0,1)$. We combine the mentioned methods and prove that the mean magnetization in our model can be positive for almost all configurations $\gamma$, and hence the quenched Gibbs states can be multiple, if the particle density $\lambda$ and the inverse temperature $\beta = 1/T$ are large enough¹. Finally, let us mention that, for our model with $\sigma _x \in \mathbb {R}$, the problem of uniqueness of Gibbs states remains open, see Sect. 2.3 below. We also note that the method developed in this article can be used to study annealed states of amorphous ferromagnetic substances where the spin and the particle configurations are in thermal equilibrium.

1.2 The Overview of the Results

In the sequel, by $\pi _\lambda$ we denote the homogeneous Poisson measure with density $\lambda >0$ – the probability distribution of the configurations $\gamma$. In Proposition 4 and Theorem 1 below, we show that there exists a set of configurations $A_1$ such that: (a) $\pi _\lambda (A_1) =1$ for all $\lambda >0$; (b) $\mathcal {G}_\mathrm{t} (\beta |\gamma ) \ne \emptyset$ for all $\beta >0$ and all $\gamma \in A_1$. Here $\mathcal {G}_\mathrm{t} (\beta |\gamma )$ is the set of tempered Gibbs measures of our spin system on $\gamma$ at a given $\beta$. Tempered Gibbs measures are Gibbs measures supported on the configurations with tempered growth of $|\sigma _x|$ as $|x| \rightarrow +\infty$. Note that, for spin models on graphs with unbounded vertex degrees and with single-spin distributions with noncompact support, there may exist states supported on configurations of spins with rapidly increasing $|\sigma _x|$, whereas for typical ferromagnetic configurations in physical substances, most of the spins take values close to same $s>0$. The proof of Theorem 1 is based on [26, Theorem 3.1] and on the property of $\pi _\lambda$ obtained in Proposition 4. Here $A_1$ appears as the set of all those configurations for which the quantities in (11) are finite, and hence the graph $(\gamma , \varepsilon _\gamma )$ belongs to the class of ‘sparse’ graphs introduced in [26].

The Ising model on $\gamma$ is a particular case of our model, which corresponds to the choice $\chi (d \sigma ) = \delta (\sigma ^2 - 1) d \sigma$. As is well-known, the set of Gibbs states of this model, $\mathcal {G}^\mathrm{Ising}_\mathrm{t} (\beta |\gamma )$, is nonempty for all $\gamma$. Next, for

$$\begin{aligned} A_2 (\beta )&:= \{ \gamma : |\mathcal {G}_\mathrm{t} (\beta |\gamma )|>1\}, \\ A^\mathrm{Ising}_2 (\beta )&:= \{ \gamma : |\mathcal {G}^\mathrm{Ising}_\mathrm{t} (\beta |\gamma )|>1\}, \end{aligned}$$

by the Wells correlation inequality [42], it follows that, cf. Proposition 5 below,

$$\begin{aligned} A_2 (\beta ) \supseteq A^\mathrm{Ising}_2 (a^2\beta ) \end{aligned}$$

(1)

where $a>0$ is determined by the measure $\chi$, see (27). For the reader convenience, we present here a complete proof of the Wells inequality, which is a refinement of that in [10, Appendix]. By standard results on the continuum percolation driven by the Poisson random point process, see [31, 33] and also [17, Corollary 3.7] and [18, Theorem 3.1], it follows that, for $\pi _\lambda$-almost all $\gamma$, the corresponding graph $(\gamma , \varepsilon _\gamma )$ has an infinite connected component whenever $\lambda > \lambda _*$, where a non-random parameter $\lambda _*>0$ is determined by the parameter $r_*$ and the dimension of the space $d$. Suppose now that each edge of this infinite connected component is removed independently with probability $1-q$ and kept with probability $q$. If the graph obtained in this way still possesses an infinite connected component, then one says that the Bernoulli bond percolation with bond probability $q$ occurs on the infinite connected component of $(\gamma ,\varepsilon _\gamma )$. As in the previous result, it is possible to show, see Propositions 2 and 3 below, that for $\lambda > \lambda _*$ there exists $q_*\in (0,1)$ such that

$$\begin{aligned} \pi _\lambda (A_3 (q)) =1, \qquad \mathrm{for} \ \ q> q_* \ \ \mathrm{and} \ \ \lambda > \lambda _*, \end{aligned}$$

(2)

where $A_3 (q)$ is the set of all those $\gamma$ such that the Bernoulli bond percolation with bond probability $q$ occurs on the infinite connected component of $(\gamma ,\varepsilon _\gamma )$. By [23, Theorem 2.1], we know that

$$\begin{aligned} A^\mathrm{Ising}_2 (\beta ) \supseteq A_3 (q), \quad \mathrm{for} \ \ \beta > [\log (1+ q) - \log (1- q)]/2, \end{aligned}$$

Then combining (1) and (2), we conclude that, for $\lambda > \lambda _*$, there exists $\beta _* \ge [\log (1+ q_*) - \log (1- q_*)]/2$ such that, for all $\beta > a^2 \beta _*$ and $\pi _\lambda$-almost all $\gamma$, the set $\mathcal {G}_\mathrm{t} (\beta |\gamma )$ contains at least two elements, see Theorem 2 below. In principle, we could stop here. However, in that case one important aspect of the theory would have been omitted. This is the dependence of our tempered Gibbs measures on $\gamma$. In mathematical theories of random systems [9], Gibbs measures are supposed to depend on the random parameters in a measurable way. Then they are called random Gibbs measures. In Sect. 4, we study the measurability issue by employing marked configurations $\hat{\gamma }$ consisting of pairs $(x, \sigma _x)$. In this setting, random Gibbs measures are obtained as conditional measures on the space of marked configurations. In Theorem 3, we show that the random Gibbs measures of our model are multiple if the conditions of Theorem 2 are satisfied.

For the sake of clarity, in this paper we restricted ourselves to the simplest model of amorphous substances—the Gilbert graph model based on the Poisson point process. In a similar way, one can prove the statements mentioned above if the underlying graph is as in the random connection model, see [18, 31, 33] or a tempered Gibbs random field, see [17, Corollary 3.7]. The only condition is that the graph almost surely has the summability property as in Proposition 4, see [14] for more detail.

2 Quenched Gibbs Measures

2.1 The Underlying Graph

Let $\mathcal {B} (\mathbb {R}^d)$ and $\mathcal {B}_\mathrm{b} (\mathbb {R}^d)$ stand for the set of all Borel and all bounded Borel subsets of $\mathbb {R}^d$, respectively. The space of all configurations is defined as

$$\begin{aligned} \Gamma =\left\{ \gamma \subset \mathbb R^d : |\gamma \cap \Lambda |<\infty \quad \mathrm{for}\mathrm{\ any} \ \Lambda \in \mathcal {B}_\mathrm{b}( \mathbb {R}^d) \right\} , \end{aligned}$$

(3)

where $|A|$ stands for the cardinality of $A$. The space $\Gamma$ is equipped with the vague topology being the weakest one in which the maps $\Gamma \ni \gamma \mapsto \sum _{x\in \gamma } f(x)$ are continuous for all continuous functions $f:\mathbb {R}^d \rightarrow \mathbb {R}$ with compact support, see, e.g., [3]. By $\mathcal {B}(\Gamma )$ we denote the corresponding Borel $\sigma$-field. The vague topology is metrizable in such a way that the corresponding metric space is complete and separable. By $\pi _\lambda$ we denote the homogeneous Poisson measure on $(\Gamma , \mathcal {B}(\Gamma ))$ with intensity (density) $\lambda >0$. Note that the set of all finite configurations $\Gamma _0$ is a Borel subset of $\Gamma$ such that $\pi _\lambda (\Gamma _0)=0$. That is, $\pi _\lambda$-almost all configurations $\gamma \in \Gamma$ are infinite.

Each $\gamma$ can be considered as a graph with vertex set $\gamma$ and adjacency relation $x \sim y$ defined by the condition $|x-y|\le r_*$. Then $\varepsilon _\gamma = \{ \{x,y\} \subset \gamma : x \sim y\}$ is its edge set. The probability distribution of the random graph $(\gamma ,\varepsilon _\gamma )$ is constructed in the following way, cf. [17]. Let $\varepsilon$ denote a set of pairs of distinct points, i.e., of $e=\{x,y\}$, $x, y \in \mathbb {R}^d$, $x\ne y$. We say that $\varepsilon$ is locally finite if $\varepsilon _\Lambda$ is finite for each $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$. Here $\varepsilon _\Lambda := \{ \{x,y\} \in \varepsilon : \{x,y\} \subset \Lambda \}$. Let $E$ be the set of all locally finite $\varepsilon$, and $\mathcal {F}(E)$ be the $\sigma$-field of subsets of $E$ generated by the counting maps $\varepsilon \mapsto |\varepsilon _\Lambda |$ with all possible $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$. Note that $E$ is, in fact, the set of locally finite configurations, cf. (3), over the symmetrization of the set $\mathbb {R}^d \times \mathbb {R}^d {\setminus } \{ (x,x): x \in \mathbb {R}^d\}$. Let $\mathcal {P}(E)$ be the set of all probability measures on $(E, \mathcal {F}(E))$. Each $\varsigma \in \mathcal {P}(E)$ can uniquely be determined by its Laplace transform, which we introduce in the following way. Let $\theta : \mathbb {R}^d \times \mathbb {R}^d \rightarrow (-1,0]$ be measurable, symmetric, and local, i.e., $\theta (x,y) = 0$ whenever $x$ or $y$ is in $\Lambda ^c:= \mathbb {R}{\setminus } \Lambda$ for some $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$. By $\Theta$ we denote the set of all such functions. Then for a $\varsigma \in \mathcal {P}(E)$, its Laplace transform is defined as

$$\begin{aligned} L_\varsigma (\theta ) = \int _{E} \exp \left[ \sum _{\{x,y\}\in \varepsilon } \log (1 + \theta (x,y)) \right] \varsigma (d \varepsilon ), \qquad \theta \in \Theta . \end{aligned}$$

Now let $g: \mathbb {R}^d \times \mathbb {R}^d \rightarrow [0,1]$ be measurable and symmetric. For each $\theta \in \Theta$, the pointwise product $g \theta$ is also in $\Theta$. An independent $g$-thinning of a given $\varsigma \in \mathcal {P}(E)$, cf. [15, Sect. 11.2], is the measure $\varsigma ^g$ defined by the relation

$$\begin{aligned} L_{\varsigma ^g} (\theta ) = L_\varsigma (g \theta ). \end{aligned}$$

(4)

The $g$-thinning of $\varsigma$ means that each configuration $\varepsilon$ distributed according to $\varsigma$ is ‘thinned’ in the sense that each $\{x,y\}\in \varepsilon$ is removed from the edge configuration with probability $1-g(x,y)$ and kept with probability $g(x,y)$. The probability distribution of such ‘thinned’ configurations is then $\varsigma ^g$.

Given $\gamma \in \Gamma$, we define $\varsigma (\cdot |\gamma )\in \mathcal {P}(E)$ by its Laplace transform

$$\begin{aligned} L(\theta |\gamma ) = \exp \left[ \sum _{x\in \gamma }\sum _{y\in \gamma {\setminus } x} \log (1 + j(x,y) \theta (x,y)) \right] \end{aligned}$$

(5)

where $j(x,y)= 1$ for $|x-y|\le r^*$ and $j(x,y)= 0$ otherwise. For a measurable $\Psi : \mathbb {R}^d \times \mathbb {R}^d \rightarrow \mathbb {R}_{+}:=[0,+\infty )$, the function

$$\begin{aligned} \Gamma \ni \gamma \mapsto \sum _{x \in \gamma } \sum _{y \in \gamma {\setminus } x} \Psi (x,y) \in \mathbb {R} \end{aligned}$$

is measurable. Therefore, the map $\Gamma \ni \gamma \mapsto L (\theta |\gamma )\in \mathbb {R}$ is also measurable for each $\theta$. Thus, $\varsigma$ defined in (5) is a probability kernel from $(\Gamma , \mathcal {B}(\Gamma ))$ to $(E, \mathcal {F}(E))$, cf. [17, Lemma 2.4]. The probability distribution of the graph $(\gamma ,\varepsilon _\gamma )$ is now defined by the measure

$$\begin{aligned} \zeta _{\lambda } (d \gamma , d \varepsilon ) = \varsigma ( d\varepsilon |\gamma )\pi _\lambda (d\gamma ). \end{aligned}$$

(6)

It may happen that $\zeta _{\lambda }$-typical graphs $(\gamma , \varepsilon _\gamma )$ have only finite connected components. Given $g$ as in (4), by $\varsigma ^g(\cdot |\gamma )$ we denote the independent $g$-thinning of $\varsigma (\cdot |\gamma )$, and set

$$\begin{aligned} \zeta ^g_{\lambda } (d \gamma , d \varepsilon ) := \varsigma ^g ( d\varepsilon |\gamma )\pi _\lambda (d\gamma ) . \end{aligned}$$

The following fact is known, see [17, Lemma 2.4 and Corollary 3.7] and especially [18, Theorem 3.1].

Proposition 1

Let $g: \mathbb {R}^d \times \mathbb {R}^d \rightarrow [0,1]$ be measurable, symmetric, and translation invariant. Then there exists $c(d)>0$ such that, for $\lambda$ satisfying the condidion

$$\begin{aligned} \lambda \int _{\mathbb {R}^d} g(x,0) d x > c(d), \end{aligned}$$

(7)

$\zeta ^g_{\lambda }$-almost all graphs $(\gamma , \varepsilon )$ have infinite connected components.

Then by Proposition 1 we obtain the following fact.

Proposition 2

There exists $\lambda _*$ such that, for $\lambda > \lambda _*$, the graph $(\gamma , \varepsilon _\gamma )$ has an infinite connected component for $\pi _\lambda$-almost all $\gamma$.

We prove Proposition 2 by taking in (7) $g(x,y) = j(x,y)$ and

$$\begin{aligned} \lambda _* = c(d)/ V(r^*), \end{aligned}$$

(8)

where $V(r^*)$ is the volume of the ball $\{x\in \mathbb {R}^d: |x| \le r^*\}$. Note that there can only be a single infinite connected component, see [31, Theorem 6.3, page 172].

For a constant function $q(x,y) \equiv q\in [0,1]$, let us consider the independent $q$-thinning of $\varsigma (\cdot |\gamma )$. If the corresponding random graph has an infinite connected component, then the Bernoulli bond percolation with bond probability $q$ occurs on the infinite connected component of $(\gamma , \varepsilon _\gamma )$. The next fact can also be deduced from Proposition 1.

Proposition 3

Let $\lambda _*$ be as in Proposition 2 and inequality $\lambda >\lambda _*$ hold. Then, there exists $q_* \in (0,1)$ such that for $\pi _\lambda$-almost all $\gamma$, the Bernoulli bond percolation with bond probability $q >q_*$ occurs on the infinite connected component of $(\gamma , \varepsilon _\gamma )$.

Indeed, by (7) and (8) one can take

$$\begin{aligned} q_* = \frac{\lambda _*}{\lambda } = \frac{c(d)}{\lambda V(r^*)}. \end{aligned}$$

Our next step is to prove that, for $\pi _\lambda$-almost all $\gamma$, the graph $(\gamma , \varepsilon _\gamma )$ belongs to the class of ‘sparse’ graphs studied in [26]. For $x\in \gamma$, let $n_\gamma (x)$ be the number of neighbors of $x$ in $\gamma$, i.e., $n_\gamma (x) := |\{y \in \gamma : y \sim x\}|$. Clearly, $n_\gamma (x)$ is finite for all $\gamma \in \Gamma$. Note, however, that

$$\begin{aligned} \sup _{x\in \gamma }n_\gamma (x)= +\infty , \end{aligned}$$

(9)

also for $\pi _\lambda$-almost all $\gamma$. Set

$$\begin{aligned} w_\alpha (x) = \exp (- \alpha |x|), \qquad \alpha >0, \ \ x \in \mathbb {R}^d. \end{aligned}$$

(10)

For $x\in \gamma$ and $\theta >0$, we then consider, cf. Eqs. (4) and (5) in [26],

$$\begin{aligned}&a_\gamma (\alpha , \theta ):= \sum _{\{x,y\} \in \varepsilon _\gamma } [w_\alpha (x) + w_\alpha (y)] [n_\gamma (x) n_\gamma (y)]^\theta ,\nonumber \\&\qquad \qquad \qquad b_\gamma (\alpha ) := \sum _{x\in \gamma } w_\alpha (x), \end{aligned}$$

(11)

and

$$\begin{aligned} A_{1,a}&= \{ \gamma \in \Gamma : \forall \alpha > 0 \ \forall \theta >0\ \ a_\gamma (\alpha , \theta ) < + \infty \} \\ A_{1,b}&= \{ \gamma \in \Gamma : \forall \alpha > 0 \ \ b_\gamma (\alpha ) < +\infty \} \\ A_1&= A_{1,a} \cap A_{1,b}. \end{aligned}$$

Proposition 4

For all $\lambda >0$, it follows that $A_1 \in \mathcal {B}(\Gamma )$ and $\pi _\lambda (A_1) =1$.

Proof

For each $\gamma$, by (11) we have that $b_\gamma (\alpha ) \le b_\gamma (\alpha ')$ whenever $\alpha > \alpha '$. Likewise, $a_\gamma (\alpha , \theta )$ is decreasing in $\alpha$ and increasing in $\theta$. This yields that

$$\begin{aligned}&A_{1,a} = \bigcap _{\alpha , \theta \in \mathbb {Q}_{+}} \{ \gamma \in \Gamma : a_\gamma (\alpha , \theta ) < + \infty \}, \\&\,A_{1,b} = \bigcap _{\alpha \in \mathbb {Q}_{+}} \{ \gamma \in \Gamma : b_\gamma (\alpha ) < + \infty \} , \end{aligned}$$

where $\mathbb {Q}_{+}$ is the set of all positive rational numbers. Hence, it is enough to obtain the $\pi _\lambda$-a.s. finiteness of $a_\gamma (\alpha , \theta )$ and $b_\gamma (\alpha )$ for fixed $\alpha \in \mathbb {Q}_{+}$ and $\theta \in \mathbb {Q}_{+}$.

By the definition of the Poisson measure $\pi _\lambda$, for each $n\in \mathbb {N}$ and any measurable and symmetric function $f:\mathbb {R}^d \times \Gamma \rightarrow \mathbb {R}_+$, we have that

$$\begin{aligned}&\int _\Gamma \left( \sum _{x\in \gamma } f(x , \gamma {\setminus } x) \right) \pi _\lambda (d \gamma ) = \lambda \int _{\Gamma } \left( \int _{\mathbb {R}^d} f(x ,\gamma ) dx \right) \pi _\lambda (d \gamma ) \end{aligned}$$

(12)

- the Mecke identity. Then

$$\begin{aligned} \int _{\Gamma } b_\gamma (\alpha ) \pi _\lambda (d \gamma ) = \int _\Gamma \left( \sum _{x\in \gamma } w_\alpha (x) \right) \pi _\lambda (d \gamma ) = \lambda \int _{\mathbb {R}^d} w_\alpha (x) d x < \infty , \end{aligned}$$

which yields

$$\begin{aligned} \pi _\lambda (\{ \gamma : b_\gamma (\alpha ) + \infty \}) =0. \end{aligned}$$

Next, we rewrite (11) in the form

$$\begin{aligned} a_\gamma (\alpha , \theta )= \sum _{x\in \gamma } w_\alpha (x) m_\gamma (\theta , x), \qquad m_\gamma (\theta , x) := \sum _{y: y\sim x}[n_\gamma (x) n_\gamma (y)]^\theta . \end{aligned}$$

(13)

Set $B_r(x) = \{y\in \mathbb {R}^d: |x-y|\le r\}$, $r>0$, and let $\mathcal {I}:\mathbb {R}^d \rightarrow \{0,1\}$ be the indicator of the ball $B_{2 r_*}(0)$. Clearly,

$$\begin{aligned} \max _{y\in B_{r_*} (x)\cap \gamma } n_\gamma (y) \le \sum _{z\in \gamma } \mathcal {I}(z-x). \end{aligned}$$

Applying this in (13) we get

$$\begin{aligned} m_\gamma (\theta , x) \le [n_\gamma (x)]^{\theta +1} \max _{y: y \sim x} [n_\gamma (y)]^\theta \le \left( \sum _{y\in \gamma {\setminus } x} \mathcal {I}(y-x)\right) ^{2\theta +1}. \end{aligned}$$

By the latter and (12), as well as by the translation invariance of $\pi _\lambda$, we then obtain from (13)

$$\begin{aligned} \int _{\Gamma } a_\gamma (\alpha , \theta )\pi _\lambda (d \gamma )&\le \int _\Gamma \sum _{x\in \gamma } w_\alpha (x) \left( \sum _{y \in \gamma {\setminus } x}\mathcal {I}(y-x) \right) ^{2\theta +1} \pi _\lambda (d\gamma )\\&= \int _{\mathbb {R}^d} w_\alpha (x) \left\{ \int _\Gamma \left( \sum _{y\in \gamma } \mathcal {I}(y-x) \right) ^{2\theta +1}\pi _\lambda (d\gamma )\right\} dx\\&= \left( \int _{\mathbb {R}^d} w_\alpha (x) dx \right) \cdot \int _\Gamma \left( \sum _{y\in \gamma } \mathcal {I}(y) \right) ^{2\theta +1}\pi _\lambda (d\gamma )\\&= \ell _{2\theta +1} \left( \lambda V (2r^*) \right) \cdot \int _{\mathbb {R}^d} w_\alpha (x) dx < \infty . \end{aligned}$$

Here

$$\begin{aligned} \ell _{\vartheta } (\varkappa ) := e^{-\varkappa }\sum _{k=1}^\infty k^{\vartheta } \varkappa ^k/k!, \qquad \vartheta , \varkappa > 0, \end{aligned}$$

and $V (2r^*)= \int _{\mathbb {R}^d} \mathcal {I}(x) dx$ is the volume of the ball $B_{2 r_*}(0)$. The latter estimate leads to the conclusion

$$\begin{aligned} \pi _\lambda ( \{ \gamma : a_\gamma (\alpha , \theta ) = +\infty \}) =0, \end{aligned}$$

which completes the proof.

2.2 The Gibbs Specification

As mentioned in the Introduction, the a priori distribution of the spin of a ‘particle’ is determined by a finite symmetric measure $\chi$ on $\mathbb {R}$. We assume that, for some $u>2$ and $\varkappa >0$, the following holds

$$\begin{aligned} \int _{\mathbb {R}} \exp \left( \varkappa |t|^u \right) \chi (d t) <\infty . \end{aligned}$$

(14)

For a fixed $\gamma$, let $\mathbb {R}^\gamma$ stand for the space of all maps $\sigma : \gamma \rightarrow \mathbb {R}$. We equip it with the topology of point-wise convergence and the corresponding Borel $\sigma$-field $\mathcal {B}(\mathbb {R}^\gamma )$. Let also $\mathcal {P}(\mathbb {R}^\gamma )$ denote the set of all probability measures on $(\mathbb {R}^\gamma , \mathcal {B}(\mathbb {R}^\gamma ))$. For $\Lambda \in \mathcal {B} (\mathbb {R}^d)$ and $\gamma \in \Gamma$, we set $\gamma _\Lambda = \gamma \cap \Lambda$ and denote by $\sigma _\Lambda$ the restriction of $\sigma$ to $\gamma _\Lambda$, i.e., $\sigma _\Lambda = (\sigma _x)_{x\in \gamma _\Lambda }$. For $\sigma , \bar{\sigma } \in \mathbb {R}^\gamma$ and $\Lambda \in \mathcal {B}(\mathbb {R}^d)$, by $\sigma _\Lambda \times \bar{\sigma }_{\Lambda ^c}$ we denote the element $\sigma ' \in \mathbb {R}^\gamma$ such that $\sigma '_\Lambda = \sigma _\Lambda$ and $\sigma '_{\Lambda ^c} = \bar{\sigma }_{\Lambda ^c}$.

The magnet that we study is characterized by a ferromagnetic spin-spin interaction, which for fixed $\Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)$ and $\gamma \in \Gamma$ is described by the following relative energy functional

$$\begin{aligned} - E^{\gamma }_\Lambda (\sigma _\Lambda | \bar{\sigma }_{\Lambda ^c})= \sum _{\{x,y\} \subset \gamma _\Lambda } \phi (|x-y|)\sigma _x \sigma _y + \sum _{ x\in \gamma _\Lambda , y \in \gamma _{\Lambda ^c}} \phi (|x-y|) \sigma _x \bar{ \sigma }_y. \end{aligned}$$

(15)

Here $\phi :\mathbb {R}_{+} \rightarrow \mathbb {R}_{+}$ is a measurable and bounded function such that, $\phi (r) \ge \phi _* >0$ for $r\in [0,r_*]$ and $\phi (r) =0$ for all $r>r_*$.

For $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$, by $\mathcal {B}_\Lambda (\mathbb {R}^{\gamma })$ we denote the smallest $\sigma$-subfield of $\mathcal {B}(\mathbb {R}^{\gamma })$ which contains all sets $A=\{ \sigma \in \mathbb {R}^\gamma : \sigma _\Lambda \in A^0\}$, $A^0 \in \mathcal {B}(\mathbb {R}^{\gamma _\Lambda })$, where $\mathcal {B}(\mathbb {R}^{\gamma _\Lambda })$ is the corresponding Borel $\sigma$-field. The algebra of local sets is

$$\begin{aligned} \mathcal {B}_\mathrm{loc} (\mathbb {R}^{\gamma }) := \bigcup _{\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)}\mathcal {B}_\Lambda (\mathbb {R}^{\gamma }). \end{aligned}$$

(16)

We will use the following topology on $\mathcal {P}(\mathbb {R}^\gamma )$, see [16, Definition 4.2, p. 59].

Definition 1

The topology of local set convergence ($\mathfrak {L}$-topology for short) is the weakest topology on $\mathcal {P}(\mathbb {R}^\gamma )$ that makes the evaluation maps $\mu \mapsto \mu (A)$ continuous for all $A\in \mathcal {B}_\mathrm{loc} (\mathbb {R}^{\gamma })$.

For $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ and $\bar{\sigma }\in \mathbb {R}^\gamma$, we define

$$\begin{aligned} \Pi ^\gamma _\Lambda (A|\bar{\sigma }) = \frac{1}{Z^\gamma _\Lambda (\bar{\sigma })}\int _{\mathbb {R}^{\gamma _\Lambda }} \mathbb {I}_A ( \sigma _\Lambda \times \bar{\sigma }_{\Lambda ^c}) \exp \left( - \beta E^\gamma _\Lambda (\sigma _\Lambda | \bar{\sigma }_{\Lambda ^c})\right) \chi _{\Lambda } (d\sigma _\Lambda ), \end{aligned}$$

(17)

where $\mathbb {I}_A$ is the indicator of $A\in \mathcal {B}(\mathbb {R}^\gamma )$, $E^\gamma _\Lambda$ is as in (15), and

$$\begin{aligned} \chi _{\Lambda } (d\sigma _\Lambda ) := \bigotimes _{x\in \gamma _\Lambda } \chi (d \sigma _x),\nonumber \\ Z^\gamma _\Lambda (\bar{\sigma }):= \int _{\mathbb {R}^{|\Lambda |}}\exp \left( - \beta E^\gamma _\Lambda (\sigma _\Lambda | \bar{\sigma }_{\Lambda ^c})\right) \chi _{\Lambda } (d\sigma _\Lambda ). \end{aligned}$$

(18)

Thus, for each $A\in \mathcal {B}(\mathbb {R}^\gamma )$, $\Pi ^\gamma _\Lambda (A|\cdot )$ is $\mathcal {B}(\mathbb {R}^\gamma )$-measurable, and, for each $\bar{\sigma }\in \mathbb {R}^\gamma$, $\Pi ^\gamma _\Lambda (\cdot | \bar{\sigma })$ is a probability measure on $(\mathbb {R}^\gamma ,\mathcal {B}(\mathbb {R}^\gamma ))$. The collection of probability kernels $\{\Pi ^\gamma _\Lambda : \Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)\}$ is called the Gibbs specification of the model we consider, see [16, Chap. 2]. It enjoys the consistency property

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \Pi ^\gamma _{\Lambda _1} (A|\sigma ) \Pi ^\gamma _{\Lambda _2} (d \sigma |\bar{\sigma }) = \Pi ^\gamma _{\Lambda _2} (A|\bar{\sigma }), \end{aligned}$$

which holds for all $A\in \mathcal {B}(\mathbb {R}^\gamma )$, $\bar{\sigma }\in \mathbb {R}^\gamma$, and all $\Lambda _1 , \Lambda _2 \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)$ such that $\Lambda _1 \subset \Lambda _2$.

Definition 2

A probability measure $\mu$ on $(\mathbb {R}^\gamma ,\mathcal {B}(\mathbb {R}^\gamma ))$ is said to be a quenched Gibbs measure (for a fixed $\gamma \in \Gamma$) if it satisfies the Dobrushin–Lanford–Ruelle equation

$$\begin{aligned} \mu (A)= \int _{\mathbb {R}^\gamma } \Pi ^\gamma _{\Lambda } (A|\sigma )\mu (d \sigma ), \qquad \mathrm{for} \ \ \mathrm{all} \ \ A\in \mathcal {B}(\mathbb {R}^\gamma ). \end{aligned}$$

The set of all such measures is denoted by $\mathcal {G}(\beta |\gamma )$.

In modern equilibrium statistical mechanics, the notion of thermodynamic phase of a system of bounded spins living on a fixed graph like $\mathbb {Z}^d$ is attributed to the extreme elements of the set of corresponding Gibbs measures, see, e.g., [16, Chap. 7] or [40, Chap. III]. However, for unbounded spins, not all extreme Gibbs measures may have physical meaning. It is believed that the measures corresponding to observed thermodynamic states should be supported on spin configurations with ‘tempered growth’ see [6, 29, 37, 38] or a more recent development in [2, 28] and [1, Chap. 3]. In this approach, only tempered Gibbs measures are taken into account, and hence a phase transition is related to the existence of multiple tempered Gibbs measures². We take this approach and study quenched Gibbs measures introduced in Definition 2 with a priori prescribed support properties. We call them tempered Gibbs states. Thus, for an $\alpha >0$, we define

$$\begin{aligned} \Sigma (\alpha ):= \left\{ \sigma \in \mathbb {R}^\gamma : \sum _{x\in \gamma } |\sigma _x|^2 w_\alpha (x) < \infty \right\} , \end{aligned}$$

(19)

where the weights $w_\alpha$ are as in (10). For each fixed $\gamma \in \Gamma$, $\Sigma (\alpha )$ is a Borel subset of $\mathbb {R}^\gamma$ and its elements are called tempered configurations. Then

$$\begin{aligned} \mathcal {G}_\mathrm{t} (\beta |\gamma ):= \{ \mu \in \mathcal {G} (\beta |\gamma ): \mu (\Sigma (\alpha )) = 1\} \end{aligned}$$

(20)

is the set of tempered Gibbs states.

Theorem 1

Let the single-spin measure $\chi$ be such that (14) holds and $A_1$ be as in Proposition 4. Then, for all $\gamma \in A_1$ and all $\beta >0$, the set of Gibbs states $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ is nonempty. Moreover, for each $\gamma \in A_1$ and each positive $\vartheta$ and $\alpha$, there exists a finite $C_\gamma (\vartheta , \alpha )>0$ such that the estimate

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \exp \left( \vartheta \sum _{x\in \gamma } |\sigma _x|^2 w_\alpha (x) \right) \mu (d \sigma ) \le C_\gamma (\vartheta , \alpha ) \end{aligned}$$

(21)

holds uniformly for all $\mu \in \mathcal {G}_\mathrm{t}(\beta |\gamma )$.

Proof

The proof of all the statements of the theorem follows by Theorem 1 of [26] since all the conditions of that theorem are satisfied in view of Proposition 4 and the assumed properties of $\chi$.

Let us make some comments. The existence of Gibbs measures follows from the relative compactness of the family $\{\Pi ^\gamma _\Lambda (\cdot |\bar{\sigma }): \Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)\}$ in the $\mathfrak {L}$-topology, see Definition 1, for at least some $\bar{\sigma }\in \Sigma (\alpha )$. A typical choice of $\bar{\sigma }$, for which the compactness is proven, is $\bar{\sigma }_x = s\in \mathbb {R}$ for all $x\in \gamma$. Note that such $\bar{\sigma }$ is tempered, see (19). Then the accumulation points of the family $\{\Pi ^\gamma _\Lambda (\cdot |\bar{\sigma }): \Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)\}$ are shown to obey the Dobrushin–Lanford–Ruelle equation and to satisfy the estimate in (21), in which the constant $C_\gamma (\vartheta , \alpha )$ can be expressed explicitly in terms of the weights as in (10) and the parameters defined in (11), cf. Proposition 4 above. Thus, the accumulation points are tempered measures, and hence belong to $\mathcal {G}_\mathrm{t} (\beta |\gamma )$. Let us stress again that $\mathcal {G}_\mathrm{t} (\beta |\gamma )$ is nonempty for all those $\gamma$, for which both $a_\gamma (\alpha , \theta )$ and $b_\gamma (\alpha )$ are finite. By similar arguments, one can show that $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ is compact in the $\mathfrak {L}$-topology. Finally, let us mention that $u>2$ in (14) and $\theta$ in Proposition 4 should be such that $\theta (u-2) > 2$. Under this condition the sufficiently fast decay of the tail of $\chi$ compensates destabilizing effect of the property (9) of the underlying graph, see [26] for more detail. Since Proposition 4 holds for all $\theta >0$, then we just assume that $u>2$.

A sequence $\{\Lambda _n\}_{n\in \mathbb {N}}\subset \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ is called cofinal if $\Lambda _{n} \subset \Lambda _{n+1}$, $n\in \mathbb {N}$, and each $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ is contained in a certain $\Lambda _n$. Given $\bar{\sigma }\in \Sigma (\alpha )$, the relative compactness of the family $\{\Pi ^\gamma _\Lambda (\cdot |\bar{\sigma }): \Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)\}$ yields that there exists a cofinal sequence $\{\Lambda _n\}_{n\in \mathbb {N}}$ such that the sequence $\{\Pi ^\gamma _{\Lambda _n} (\cdot |\bar{\sigma })\}_{n\in \mathbb {N}}$ converges in the $\mathfrak {L}$-topology to a certain element of $\mathcal {G}_\mathrm{t}(\beta |\gamma )$. For $a> 0$, by

$$\begin{aligned} \mu ^{\pm a}\in \mathcal {G}_\mathrm{t}(\beta |\gamma ) \end{aligned}$$

(22)

we denote limiting elements of $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ which correspond to $\bar{\sigma }_x= \pm a$ for all $x \in \gamma$. Each such $\mu ^{\pm a}$ depends on the sequence $\{\Lambda _n\}_{n\in \mathbb {N}}$ along which it has been attained. Note that only limiting Gibbs measures can approximate thermodynamic states of large finite systems, see [16, Sect. 7.1].

Now we turn to the single-spin measure $\chi$. If it has compact support, as was the case in [36], then (14) clearly holds for any $u$ and $\varkappa$. The most known example of such $\chi$ is

$$\begin{aligned} \chi (d t) = [\delta _{-1} ( dt) + \delta _{+1} ( dt)]/2, \end{aligned}$$

(23)

which corresponds to an Ising magnet. Here $\delta _s$ is the Dirac measure concentrated at $s\in \mathbb {R}$. We reserve a special notation $\mathcal {G}^{ \mathrm {Ising}}(\beta |\gamma )$ for the set of all corresponding Gibbs measures. By $\nu ^{\pm }\in \mathcal {G}^{ \mathrm {Ising}}(\beta |\gamma )$, we denote the limiting Gibbs measures as in (22) with $a =1$. In this case, however, $\nu ^{\pm }$ are independent of the sequences $\{\Lambda _n\}_{n\in \mathbb {N}}$ along which they were attained. This holds because, for each $x$ and $\nu \in \mathcal {G}^{ \mathrm {Ising}}(\beta |\gamma )$,

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _x \nu ^{-} (d \sigma ) \le \int _{\mathbb {R}^\gamma } \sigma _x \nu (d \sigma ) \le \int _{\mathbb {R}^\gamma } \sigma _x \nu ^{+} (d \sigma ). \end{aligned}$$

That is, $\nu ^{+}$ and $\nu ^{-}$ are the maximum and minimum elements of $\mathcal {G}^{ \mathrm {Ising}}(\beta |\gamma )$, respectively, cf. [28, Theorem 3.8].

In the case of ‘unbounded’ spins, a natural choice of the single-spin measure is

$$\begin{aligned} \chi (d t) = \exp \left( - V(t)\right) dt, \end{aligned}$$

where $V: \mathbb {R} \rightarrow \mathbb {R}$ is a measurable even function such that: (a) the set $\{t\in \mathbb {R}: V(t) < +\infty \}$ is of positive Lebesgue measure; (b) $V(t)$ increases at infinity as $|t|^{u + \epsilon }$ with some $\epsilon >0$ and $u$ as in (14). This includes the case where $V$ is a polynomial of even degree at least $4$ with positive leading coefficient, cf. [26–29].

2.3 The Question of Uniqueness

Once the existence of Gibbs states has been established, the problem of their uniqueness/nonuniqueness arises. Thus, prior to proving non-uniqueness of $\mu \in \mathcal {G}_\mathrm{t}(\beta |\gamma )$, which holds for $\pi _\lambda$-almost all $\gamma$ whenever $\beta$ and $\lambda$ are large enough, see Theorem 2 below, we address the question of whether the same uniqueness does actually hold for some values of these parameters. For small enough $\lambda$, all the connected components of the graph $(\gamma , \varepsilon _\gamma )$ are finite, and hence $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ is a singleton for all $\beta$. On the other hand, $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ is a singleton if and only if, for each $x\in \gamma$, arbitrary $\bar{\sigma }\in \Sigma (\alpha )$, and any cofinal sequence $\{\Lambda _n\}_{n \in \mathbb {N}}$, one has

$$\begin{aligned} \lim _{n\rightarrow +\infty } \int _{\mathbb {R}^\gamma } \sigma _x \Pi ^\gamma _{\Lambda _n}( d\sigma |\bar{\sigma }) = 0. \end{aligned}$$

(24)

This equivalence holds for any symmetric ferromagnet satisfying the bound in (21), that can be proven by standard arguments based on the Strassen theorem, see [28] for more detail. Actually, for models with ‘unbounded’ spins living on an infinite connected graph with globally unbounded degree, which by (9) is the case in our situation, there are no tools³ for proving (24). For the Ising ferromagnet, the uniqueness in question can be obtained by percolation arguments, see [19, Theorem 7.2].

3 The Phase Transition

3.1 The Statement

Recall that by a phase transition in the considered ferromagnet we mean the fact that the set of tempered Gibbs states $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ for $\pi _\lambda$-almost all $\gamma$ contains at least two elements (if $\beta$ and $\lambda$ are big enough). It is equivalent to the appearance of a nonzero magnetization in states $\mu ^{\pm a}\in \mathcal {G}_\mathrm{t}(\beta |\gamma )$, cf. [16, Chap. 19] and (24).

Let us note that there is no interaction between spins in different connected components of the underlying graph. Then for a phase transition to occur it is necessary that the graph $(\gamma , \varepsilon _\gamma )$ possess an infinite connected component, which holds for $\pi _\lambda$-almost all $\gamma$ whenever $\lambda > \lambda _*$, see [31, 33] and also [17, Corollary 3.7] and [18, Theorem 3.1]. For $\lambda <\lambda _{*}$, we have no infinite connected component of $(\gamma , \varepsilon _\gamma )$ and thus $\left| \mathcal {G}(\beta |\gamma )\right| =1$ for all $\beta$ and $\pi _\lambda$-almost all $\gamma$. In order to obtain a sufficient condition for a phase transition to occur, we will explore the well-known relationship between the Bernoulli bond percolation on the fixed sample graph $(\gamma , \varepsilon _\gamma )$, established in Propositions 2 and 3, and the existence of multiple Gibbs states in the corresponding Ising model, established in [23]. Our goal is to prove the following statement.

Theorem 2

Let the measure $\chi$ be as in Theorem 1 and such that $\chi (\{0\}) < \chi (\mathbb {R})$. Assume also that the intensity $\lambda$ of the underlying Poisson point process satisfies the condition $\lambda >\lambda _{*}$, and thus the typical graph $(\gamma , \varepsilon _\gamma )$ has an infinite connected component. Then there exists a constant $\beta ^*>0$ such that, for $\beta > \beta ^*$ and $\pi _\lambda$-almost all $\gamma$, the sets $\mathcal {G}_\mathrm{t}(\beta |\gamma )$ contain at least two elements.

The proof of this statement is based on the following result, cf. (22).

Lemma 1

Let the conditions of Theorem 2 be satisfied. Then there exist $a>0$, and $\beta _*>0$ such that, for $\pi _\lambda$-almost all $\gamma$, and for all $\beta > a^2 \beta _*$, all $\mu ^{+a}\in \mathcal {G}(\beta |\gamma )$, and some $o \in \gamma$, the following estimate holds:

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _o \mu ^{+a}(d \sigma ) > 0. \end{aligned}$$

(25)

The proof of this lemma is given in the next subsection.

Proof of Theorem 2: Since the integral in (25) is the limit of those in (24) with $\bar{\sigma }_x = a$, then (25) contradicts (24) and hence implies non-uniqueness, which ought to hold for $\beta > \beta ^*:= a^2 \beta _*$. On the other hand, by the invariance of $\chi$ and of the interaction in (15) with respect to the transformation $\sigma \rightarrow - \sigma$ and $\bar{\sigma } \rightarrow - \bar{\sigma }$, we have

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _o \mu ^{+a}(d \sigma ) = - \int _{\mathbb {R}^\gamma } \sigma _o \mu ^{-a}(d \sigma ). \end{aligned}$$

Then (25) yields $\mu ^{+a} \ne \mu ^{-a}$ and hence the multiplicity in question. Note that $o$ in (25) belongs to the infinite connected component of $(\gamma , \varepsilon _\gamma )$, and the integral in (25) is the mean value of the spin at this vertex in state $\mu ^{+a}$.

3.2 Proof of Lemma 1

First, by means of the percolation arguments of [23], we prove the lemma for the Ising model. Then we extend the proof to the general case by comparison inequalities.

Recall that the single-spin measure of the Ising model is given in (23), $\mathcal {G}^{ \mathrm {Ising}}(\beta |\gamma )$ denotes the set of all corresponding Gibbs measures, and $\nu ^+ \in \mathcal {G}^\mathrm{Ising}(\beta |\gamma )$ is the maximum Gibbs measure as in (22) with $a=1$. We are going to use the key fact proven in [23]: the Ising model with constant intensities $\phi (|x-y|) = \phi _*>0$ on the edges of an infinite graph has at least two phases if and only if the graph admits the Bernoulli bond percolation with critical probability $q_* \in (0,1)$ if $\beta > [\log (1+q_*) - \log (1-q_*)]/2\phi _*$. In our case, for $\pi _\lambda$-almost all $\gamma$, the graph $(\gamma , \varepsilon _\gamma )$ admits this percolation and the threshold probability satisfies $q_* \ge \lambda _*/\lambda$, see Proposition 3. Then, for some $o\in \gamma$, it follows that

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _o \tilde{\nu }^{+} (d \sigma ) >0, \end{aligned}$$

(26)

see [23, Theorem 2.1] and also the proof of Lemma 4.2 therein. Here $\tilde{\nu }^{+}$ is the corresponding Gibbs measure of the Ising model with $\phi (|x-y|) = \phi _*>0$. By the standard GKS inequality, see, e.g., [23, Sect. 3.4], we have

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _o \nu ^{+} (d \sigma ) \ge \int _{\mathbb {R}^\gamma } \sigma _o \tilde{\nu }^{+} (d \sigma ), \end{aligned}$$

which together with (26) yields the proof in this case.

Now we turn to the general case and estimate the integral in (25) from below by the corresponding integral with respect to the maximum Gibbs measure $\nu ^{+}$ of the Ising model with a rescaled interaction intensity. The proof of the lemma immediately follows from the Wells inequality used, e.g., in [35].

Proposition 5

(Wells inequality) Let $a>0$ be such that

$$\begin{aligned} \chi \left( [a \sqrt{2}, +\infty )\right) \ge \chi ([0,a]). \end{aligned}$$

(27)

Then, for each $x\in \gamma$ and each $\mu ^{+a} \in \mathcal {G}(\beta |\gamma )$, as well as for $\nu ^{+} \in \mathcal {G}^\mathrm{Ising}(a^2\beta |\gamma )$, we have that

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _x \mu ^{+a} (d\sigma ) \ge a \int _{\mathbb {R}^\gamma } \sigma _x \nu ^{+} (d\sigma ) . \end{aligned}$$

(28)

As the original publication [42] is hardly attainable and the proof in [10] contains numerical inaccuracies, for the reader convenience in Appendix we give a short proof of this inequality in the form suitable for our purposes.

4 Random Gibbs Measures

Random Gibbs measures of spin systems are supposed to depend on the random parameters ($\gamma$ in ours case) in a measurable way, see, e.g., [9, Definition 6.2.5] and [27, Definition 2.3]. At the same time, the measures introduced in Definition 2 and (20) are defined on spaces which themselves depend on $\gamma$, and thus one cannot speak of the corresponding measurability in this setting. In order to settle this problem, we use some of the results of [14] where random Gibbs measures are defined as conditional measures on spaces of marked configurations. Here we outline the main points of this construction and prove the non-uniqueness of such measures, see Theorem 3 below.

For our model, the space of marked configurations is

$$\begin{aligned} \widehat{\Gamma } = \{ \hat{\gamma }= ( \gamma , \sigma ): \gamma \in \Gamma , \ \sigma \in \mathbb {R}^\gamma \}, \end{aligned}$$

where $\Gamma$ is as in (3). Thus, the elements of $\widehat{\Gamma }$ consist of pairs $(x,\sigma _x)$, and $\hat{\gamma } = \hat{\gamma }'$ implies $\gamma = \gamma '$. To relate $\hat{\gamma }$ with $\gamma$ we use the canonical projection

$$\begin{aligned} \widehat{\Gamma } \ni \hat{\gamma } \mapsto \hat{p}(\hat{\gamma }) =\gamma \in \Gamma , \end{aligned}$$

(29)

and equip $\widehat{\Gamma }$ with the following topology. Let $f: \mathbb {R}^d \times \mathbb {R}\rightarrow \mathbb {R}$ be a bounded continuous function with support contained in $\Lambda \times \mathbb {R}$ for some $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$. The topology in question is the weakest one which makes the maps

$$\begin{aligned} \widehat{\Gamma } \ni \hat{\gamma } \mapsto \sum _{x \in \hat{p}(\hat{\gamma })} f(x, \sigma _x) \in \mathbb {R} \end{aligned}$$

(30)

continuous for all possible $f: \mathbb {R}^d \times \mathbb {R}\rightarrow \mathbb {R}$ as above. This topology is completely and separably metrizable, see [13, Sect. 2], and thus $\widehat{\Gamma }$ is a Polish space. Let $\mathcal {B}(\widehat{\Gamma })$ denote the corresponding Borel $\sigma$-field. From the definition of the topologies of $\Gamma$ and $\widehat{\Gamma }$, it follows that the projection defined in (29) is continuous, and hence, for each $\gamma \in \Gamma$, we have that

$$\begin{aligned} \hat{p}^{-1}(\{\gamma \})=: \mathbb {R}^\gamma \in \mathcal {B}(\widehat{\Gamma }). \end{aligned}$$

For each fixed $\gamma \in \Gamma$, $\mathbb {R}^\gamma$ is a Polish space embedded into $\widehat{\Gamma }$, which is a Polish space as well. By the Kuratowski theorem [39, page 21], the latter implies that the Borel $\sigma$-fields $\mathcal {B}(\mathbb {R}^\gamma )$ and

$$\begin{aligned} \mathcal {A}( \mathbb {R}^\gamma ) := \{ A \in \mathcal {B}(\widehat{\Gamma }): A \subset \mathbb {R}^\gamma \} \end{aligned}$$

are measurably isomorphic. Thus, any probability measure $\mu$ on $\mathcal {B}(\widehat{\Gamma })$ with the property $\mu (\mathbb {R}^\gamma )=1$ can be redefined as a measure on $\mathcal {B}(\mathbb {R}^\gamma )$.

Let $\mathcal {P}(\widehat{\Gamma })$ be the set of all probability measures on $(\widehat{\Gamma }, \mathcal {B}( \widehat{\Gamma }))$. We equip it with the topology defined as follows. For a fixed $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$, let $\mathcal {B}_\Lambda (\widehat{\Gamma })$ be the smallest $\sigma$-subfield of $\mathcal {B} (\widehat{\Gamma })$ such that the maps as in (30) are $\mathcal {B}_\Lambda (\widehat{\Gamma })$-measurable for all bounded measurable functions $f: \mathbb {R}^d \times \mathbb {R} \rightarrow \mathbb {R}$ with support contained in $\Lambda \times \mathbb {R}$. Then we set, cf. (16),

$$\begin{aligned} \mathcal {B}_\mathrm{loc}(\widehat{\Gamma }) = \bigcup _{ \Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)} \mathcal {B}_{\Lambda } (\widehat{\Gamma }). \end{aligned}$$

Now the $\mathfrak {L}$-topology on $\mathcal {P}(\widehat{\Gamma })$ is defined as in Definition 1 by using $\mathcal {B}_\mathrm{loc}(\widehat{\Gamma })$ as the algebra of local sets. A map $\widehat{\Gamma } \ni \hat{\gamma } \mapsto \varphi (\hat{\gamma })\in \mathbb {R}$ is called local if it is $\mathcal {B}_\Lambda (\widehat{\Gamma })$-measurable for some $\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$. Local maps $\Gamma \ni \gamma \mapsto \varphi (\gamma )\in \mathbb {R}$ are defined in the same way.

Let $\bar{\sigma }$ in (17) be fixed in such a way that $\sigma _x = s$ for some $s\in \mathbb {R}$ and all $x\in \gamma$. Then, for each $\gamma \in \Gamma$ and $\Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)$ and for such $\bar{\sigma }$, we can define a probability measure on $(\widehat{\Gamma }, \mathcal {B}( \widehat{\Gamma }))$ by setting

$$\begin{aligned} \widehat{\Pi }_\Lambda ^s(A|\gamma ) = \Pi _\Lambda ^\gamma (A \cap \mathbb {R}^\gamma |\bar{\sigma }), \end{aligned}$$

(31)

where $\Pi _\Lambda ^\gamma (\cdot |\bar{\sigma })$ is given in (17). By the very construction, the map $\Gamma \ni \gamma \mapsto \widehat{\Pi }_\Lambda ^s (A|\gamma )\in \mathbb {R}$ is measurable for each $A\in \mathcal {B}(\widehat{\Gamma })$. Thus, we can set

$$\begin{aligned} \hat{\eta }_\Lambda ^s (\cdot ):= \int _{\Gamma } \widehat{\Pi }_\Lambda ^s (\cdot |\gamma )\pi _\lambda (d \gamma ), \end{aligned}$$

(32)

which is an element of $\mathcal {P}(\widehat{\Gamma })$, equipped with the $\mathfrak {L}$-topology defined above. It can be shown, see [14, Corollary 4.2], that, for each $s\in \mathbb {R}$, the family $\{\hat{\eta }^s_\Lambda \}_{\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)}$ is relatively compact in $\mathcal {P}(\widehat{\Gamma })$ . Let $\hat{\eta }^s$ be its accumulation point and $\{\Lambda _n\}_{n\in \mathbb {N}}$ be the cofinal sequence such that $\hat{\eta }^s_{\Lambda _n} \rightarrow \hat{\eta }^s$ as $n \rightarrow +\infty$. By $\hat{\eta }_\Gamma ^s$ we denote the projection of $\hat{\eta }^s$ on $\Gamma$. For a bounded local function $f:\Gamma \rightarrow \mathbb {R}$, $\hat{f}:=f\circ \hat{p}:\widehat{\Gamma } \rightarrow \mathbb {R}$ is then also local, and hence

$$\begin{aligned} \int _\Gamma f d \pi _\lambda = \int _{\widehat{\Gamma }} \hat{f} d \hat{\eta }^s_{\Lambda _n} \rightarrow \int _{\widehat{\Gamma }} \hat{f} d \hat{\eta }^s = \int _\Gamma f d \hat{\eta }^s_\Gamma , \quad n \rightarrow +\infty . \end{aligned}$$

Thus, $\hat{\eta }^s_\Gamma = \pi _\lambda$, and we can disintegrate, cf. (32), and obtain

$$\begin{aligned} \hat{\eta }^s (A) = \int _{\Gamma }\eta ^s (A|\gamma ) \pi _\lambda (d \gamma ), \quad A\in \mathcal {B}(\widehat{\Gamma }), \end{aligned}$$

(33)

where $\eta ^s$ is a regular conditional measure such that $\eta ^s (A|\gamma ) = \eta ^s (A\cap \mathbb {R}^\gamma |\gamma )$ for almost all $\gamma$. As in (31), we then redefine $\eta ^s(\cdot |\gamma )$ as a measure on $\mathbb {R}^\gamma$, for which we keep the same notation. One can prove [14] that, for almost all $\gamma$, $\eta ^s(\cdot |\gamma ) \in \mathcal {G}_\mathrm{t}(\beta |\gamma )$. Thus, $\eta ^s(\cdot |\gamma )$ is a random Gibbs measure.

In principle, we could construct such Gibbs measures without the study performed in Sect. 2, just by showing that the family $\{\hat{\eta }^s_\Lambda \}_{\Lambda \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)}$ is relatively compact in $\mathcal {P}(\widehat{\Gamma })$. However, this way has the following drawbacks: (a) in contrast to those in (22), the measures $\eta ^s(\cdot |\gamma )$ need not be limiting and hence cannot approximate thermodynamic states of large finite systems; (b) there is no control on the sets of $\gamma$, as well as on their dependence on $\lambda$ and $s$, for which $\eta ^s(\cdot |\gamma )$ exist, cf. Proposition 4; (c) nothing can be said of the integrability properties of $\eta ^s(\cdot |\gamma )$, cf. (21); (d) it is unclear whether we can have $\eta ^{+a} \ne \eta ^{-a}$. These problems are partly resolved in the statement below. Recall that each $\mu ^{+a}$ is a measure on $\mathbb {R}^\gamma$, $\gamma \in A_1$, see Proposition 4, and is attained along a cofinal sequence $\mathcal {L}:= \{\Lambda _n\}_{n\in \mathbb {N}}$, that will be indicated as $\mu ^{+a}_{\mathcal {L}}$. For each $\gamma \in A_1$, such measures $\mu ^{+a}$ constitute the set of accumulation points of the family $\{\Pi ^\gamma _\Lambda (\cdot |\bar{\sigma }): \Lambda \in \mathcal {B}_\mathrm{b} (\mathbb {R}^d)\}$. The meaning of the theorem below is that, for a full $\pi _\lambda$-measure subset $A_1' \subset A_1$, there exists a (measurable) selection $\{\mu ^{+a}_{\mathcal {L}(\gamma )}\}_{\gamma \in A_1'}$ such that $\mu ^{+a}_{\mathcal {L}(\gamma )} = \eta ^{+a}(\cdot |\gamma )$ for all $\gamma \in A_1'$.

Theorem 3

For arbitrary positive $a$ and $\lambda$, there exists $A_1' \subset A_1$ such that: (i) $\pi _\lambda (A_1') =1$; (ii) for each $\gamma \in A_1'$, there exists a cofinal sequence $\mathcal {L}(\gamma )$ such that $\mu ^{+a}_{\mathcal {L}(\gamma )} = \eta ^{+a}(\cdot |\gamma )$. Therefore, $\eta ^{+a}(\cdot |\gamma ) \ne \eta ^{-a}(\cdot |\gamma )$, and hence quenched random Gibbs measures are multiple whenever the conditions of Theorem 2 are satisfied.

Proof

From now on we fix $s = +a$ and $\{\Lambda _n\}_{n\in \mathbb {N}}$, and hence $\hat{\eta }^s$. Note that we cannot expect that the assumed convergence $\hat{\eta }^s_{\Lambda _n} \rightarrow \hat{\eta }^s$ does imply that $\Pi ^\gamma _{\Lambda _n} (\cdot |\bar{\sigma }) \rightarrow \eta ^s (\cdot |\gamma )$, which would yield the proof.

Let $\{\Xi _n\}_{n\in \mathbb {N}}\subset \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ be a partition of $\mathbb {R}^d$. Fix dense subsets $\{y^n_k\}_{k\in \mathbb {N}} \subset \Xi _n$. As in the proof of Lemma 2.3 in [24, p. 20], by means of this partition we introduce a linear order of the elements of each $\gamma \in \Gamma$. If $x, y\in \gamma$ belong to distinct $\Xi _n$, we set $x < y$ whenever $x\in \Xi _n$, $y\in \Xi _m$, and $n<m$. If both $x$ and $y$ lie in the same $\Xi _n$, let $k$ be the smallest integer such that $|x-y^n_k | \ne |y-y^n_k |$. Then we set $x<y$ if $|x-y^n_k | <|y-y^n_k |$. Next, we enumerate the elements of $\gamma$ in accordance with the order in such a way that $x_1 < x_2 < \cdots < x_k < \cdots$. This defines an enumeration on $\widehat{\Gamma }$, that is, the map

$$\begin{aligned} \hat{\gamma } \mapsto \epsilon (\hat{\gamma }) = \{ (x_1 , \sigma _{x_1}), \dots , (x_k , \sigma _{x_k}), \dots \} \end{aligned}$$

(34)

such that all $(x_k, \sigma _{x_k})$ are distinct and $(x_k, \sigma _{x_k}) \in \hat{\gamma }$. This map is measurable in the sense that $\{ \hat{\gamma } : x_k \in \Delta , \ \sigma _{x_k} \in \Sigma \} \in \mathcal {B}(\widehat{\Gamma })$ for each $k\in \mathbb {N}$, $\Delta \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ and $\Sigma \in \mathcal {B}(\mathbb {R})$. The latter fact can be proven by a slight generalization of the proof of Lemma 2.3 in [24, p. 20]. Then, for a fixed $\Delta \in \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ and any $k\in \mathbb {N}$, $p \in (\mathbb {Q}^d)^{\mathbb {N}}$, and $q\in \mathbb {Q}^{\mathbb {N}}$, the real and imaginary parts of the function

$$\begin{aligned} \hat{\gamma } \mapsto \exp [i (p_k\cdot x_k + q_k\sigma _{x_k})]\in \mathbb {C} \end{aligned}$$

are as in (30), and hence are $\mathcal {B}_\Delta (\widehat{\Gamma })$- measurable. Here $i = \sqrt{-1}$ and $p\cdot x$ stands for the scalar product in $\mathbb {R}^d$. Let $\mathcal {D}=\{\Delta _n\}_{n\in \mathbb {N}}\subset \mathcal {B}_\mathrm{b}(\mathbb {R}^d)$ be a cofinal sequence. For $\epsilon$ as in (34), $\Delta \in \mathcal {D}$, $p \in (\mathbb {Q}^d)^{\mathbb {N}}$, and $q\in \mathbb {Q}^{\mathbb {N}}$, we set

$$\begin{aligned} f_{\Delta , p, q} (\hat{\gamma }) = \exp \left( i \sum _{j=1}^{|\gamma _\Delta |} \left[ p_{k_j}\cdot x_{k_j} + q_{k_j} \sigma _{x_{k_j}} \right] \right) , \end{aligned}$$

(35)

where $\gamma = \hat{p}(\hat{\gamma })$, $(x_k, \sigma _{x_k})$ is the $k$-th element of the sequence $\epsilon (\hat{\gamma })$, and the sum runs over the set of all those $j$ for which $x_{k_j} \in \Delta$. By construction, each such $f$ is $\mathcal {B}_\Delta (\widehat{\Gamma })$-measurable. Let $\mathcal {F}$ be the (countable) family of all such functions. It has the following properties: (a) is closed under point-wise multiplication; (b) separates points of $\widehat{\Gamma }$. The latter means that, for any two distinct $\hat{\gamma }, \hat{\gamma }' \in \widehat{\Gamma }$, one finds $f\in \mathcal {F}$ such that $f(\hat{\gamma }) \ne f(\hat{\gamma }')$. By Fernique’s theorem [41, p. 6], property (b) implies that $\sigma (\mathcal {F}) = \mathcal {B}(\widehat{\Gamma })$. That is, the smallest $\sigma$-field of subsets of $\widehat{\Gamma }$ with respect to which each $f\in \mathcal {F}$ is measurable, is $\mathcal {B}(\widehat{\Gamma })$. Combining this with property (a) we then obtain, see [7, . 149] or the proof of Theorem 1.3.26 in [1, p. 113], that $\mathcal {F}$ is a separating class for $\mathcal {P}(\widehat{\Gamma })$. That is, $\mu ,\nu \in \mathcal {P}(\widehat{\Gamma })$ coincide if and only if

$$\begin{aligned} \int f d \mu = \int f d \nu , \qquad \mathrm{for} \ \mathrm{all} \ f \in \mathcal {F}. \end{aligned}$$

For a fixed triple $\Delta , p, q$, by the assumed convergence $\hat{\eta }^s_{\Lambda _n} \rightarrow \hat{\eta }^s$ we have that

$$\begin{aligned} \int _{\widehat{\Gamma }} f_{\Delta , p, q} (\hat{\gamma } ) \hat{\eta }^s_{\Lambda _n} ( d\hat{\gamma })= \int _\Gamma g^{(n)}_{\Delta , p, q} (\gamma ) \pi _\lambda ( d \gamma ) \rightarrow \int _{\widehat{\Gamma }} f_{\Delta , p, q} (\hat{\gamma } ) \hat{\eta }^s ( d\hat{\gamma }), \end{aligned}$$

(36)

as $n \rightarrow +\infty$. Here, cf. (33) and (35),

$$\begin{aligned} g^{(n)}_{\Delta , p, q} (\gamma )&:= \int _{\widehat{\Gamma }} f_{\Delta , p, q} (\hat{\gamma } ) \widehat{\Pi }^{s}_{\Lambda _n} ( d \hat{\gamma } |\gamma ) = \exp \left( i \sum _{j=1}^{|\gamma _\Delta |} p_{k_j}\cdot x_{k_j} \right) h^{(n)}_{\Delta , q} (\gamma ),\nonumber \\ h^{(n)}_{\Delta , q} (\gamma )&:= \int _{\mathbb {R}^\gamma } \exp \left( i \sum _{x \in \gamma _\Delta } q_{k(x) } \sigma _x \right) \Pi ^\gamma _{\Lambda _n} (d \sigma |\bar{\sigma }), \end{aligned}$$

(37)

where $k(x)$ is the number of $x \in \gamma = \hat{p}(\hat{\gamma })$ defined by the enumeration (34). Obviously, $|h^{(n)}_{\Delta ,q} (\gamma )|\le 1$ for all $n\in \mathbb {N}$, and $\{ h^{(n)}_{\Delta , q} \}_{n\in \mathbb {N}} \subset L^1 (\Gamma , d \pi _\lambda )$. By Komlós’ theorem (see, e.g., [4, 27]), there exists a subsequence $\{ h^{(n_l)}_{\Delta , q} \}_{l\in \mathbb {N}} \subset \{ h^{(n)}_{\Delta , q} \}_{n\in \mathbb {N}}$ such that, for each further subsequence $\{ h^{(n_{l_m})}_{\Delta , q} \}_{m\in \mathbb {N}} \subset \{ h^{(n_l)}_{\Delta , q} \}_{l\in \mathbb {N}}$, one has

$$\begin{aligned} \frac{1}{M} \sum _{m=1}^M h^{(n_{l_m})}_{\Delta , q} (\gamma ) \rightarrow h_{\Delta , q} (\gamma ), \qquad \mathrm{for} \ \pi _\lambda -\mathrm{a.a.} \ \gamma \in \Gamma , \end{aligned}$$

(38)

where $h_{\Delta , q}$ is a certain element of $L^1 (\Gamma , d \pi _\lambda )$. Note that the subsequence $\{n_l\}_{l\in \mathbb {N}}$ depends on the choice of $\Delta , q$. However, by the diagonal procedure as in [27] one can pick $\{n_{l_m}\}_{m\in \mathbb {N}} \subset \{n_l\}_{l\in \mathbb {N}}$ such that (38) holds for all $\Delta \in \mathcal {D}$, and $q\in \mathbb {Q}^{\mathbb {N}}$. Then by (37) and (38) we get

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \exp \left( i \sum _{x \in \gamma _\Delta } q_{k(x) } \sigma _x \right) P^\gamma _{M} (d \sigma |\bar{\sigma }) \rightarrow h_{\Delta , q} (\gamma ), \quad \mathrm{for} \ \pi _\lambda -\mathrm{a.a.} \ \gamma \in \Gamma , \end{aligned}$$

(39)

where $\Lambda ^m := \Lambda _{n_{l_m}}$ and

$$\begin{aligned} P^\gamma _{M} (d \sigma |\bar{\sigma }) := \frac{1}{M} \sum _{m=1}^M \Pi ^\gamma _{\Lambda ^m} (d \sigma | \bar{\sigma }), \quad \ \ \bar{\sigma }_x = s = +a, \quad \ x \in \gamma . \end{aligned}$$

(40)

Note that by Komlós’ theorem the convergence in (39) holds also for the Cesàro means of each subsequence of $\{\Pi ^\gamma _{\Lambda ^m} (\cdot | \bar{\sigma })\}_{m\in \mathbb {N}}$. For any $\gamma \in A_1$, by Proposition 4 $\{\Pi ^\gamma _{\Lambda ^m} (\cdot | \bar{\sigma })\}_{m\in \mathbb {N}}$ is relatively compact in the $\mathfrak {L}$-topology. Thus, one can pick $\mathcal {L}(\gamma )=\{\Lambda ^{m_l}(\gamma )\}_{l\in \mathbb {N}} \subset \{\Lambda ^m\}_{m\in \mathbb {N}}$, for which, cf. (22),

$$\begin{aligned} \Pi ^\gamma _{\Lambda ^{m_l}(\gamma )} (\cdot | \bar{\sigma }) \rightarrow \mu ^{+a}_{\mathcal {L}(\gamma )}\in \mathcal {G}_\mathrm{t}(\beta |\gamma ), \quad l\rightarrow +\infty . \end{aligned}$$

(41)

Note that the dependence of the sequence $\mathcal {L}(\gamma )$ on $\gamma$ can be very irregular in view of the so called chaotic size dependence, see [32] and the discussion in [27]. Now let $A_1'\subset A_1$ be such that also (39) holds for all $\Delta$ and $q$. Then, for a fixed $\gamma \in A_1'$, by (40) and (41) there exists the sequence $\{ P^\gamma _{M} \}$ such that $P^\gamma _{M} \rightarrow \mu ^{+a}_{\mathcal {L}(\tau )}$, which yields

$$\begin{aligned} h_{\Delta , q} (\gamma ) = \int _{\mathbb {R}^\gamma } \exp \left( i \sum _{x \in \gamma _\Delta } q_{k(x) } \sigma _x \right) \mu ^{+a}_{\mathcal {L}(\gamma )}(d \sigma ), \quad \gamma \in A_1' . \end{aligned}$$

Since $h_{\Delta , q} \in L^1(\Gamma ,d \pi _\lambda )$, we can integrate and by (33), (36), (37), and (38) obtain

$$\begin{aligned}&\int _{\Gamma } \exp \left( i \sum _{j=1}^{|\gamma _\Delta |} p_{k_j}\cdot x_{k_j} \right) \left[ \,\, \int _{\mathbb {R}^\gamma } \exp \left( i \sum _{x \in \gamma _\Delta } q_{k(x) } \sigma _x \right) \eta ^{+a}(d \sigma |\gamma )\right] \pi _\lambda (d \gamma )\\&= \int _{\Gamma } \exp \left( i \sum _{j=1}^{|\gamma _\Delta |} p_{k_j}\cdot x_{k_j} \right) \left[ \,\,\int _{\mathbb {R}^\gamma } \exp \left( i \sum _{x \in \gamma _\Delta } q_{k(x) } \sigma _x \right) \mu ^{+a}_{\mathcal {L}(\gamma )}(d \sigma )\right] \pi _\lambda (d \gamma ), \end{aligned}$$

which holds for all $\Delta \in \mathcal {D}$, $p \in (\mathbb {Q}^d)^{\mathbb {N}}$, and $q\in \mathbb {Q}^{\mathbb {N}}$. This yields the proof since the integrand functions constitute separating classes.

Appendix: Proof of Proposition 5

For the general choice of $\chi$, let $\Pi ^{\gamma ,+a}_\Lambda$ be defined as in (17) with $\bar{\sigma }_x = +a$ for all $x\in \gamma$. Each $\mu ^{+a}$ is the limit of $\{\Pi _{\Lambda _n}^{\gamma ,+a}\}_{n\in \mathbb {N}}$ for some cofinal sequence $\{\Lambda _n\}_{n\in \mathbb {N}}$. In the case of unbounded spins, this convergence alone does not yet imply the convergence of the moments of $\Pi _{\Lambda _n}^{\gamma ,+a}$ to that on the left-hand side of (28). Then we use the uniform in $n$ bound as in (21), which can also be proven for all $\Pi _\Lambda ^{\gamma ,+a}$, and obtain

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _x \Pi _{\Lambda _n}^{\gamma ,+a} (d \sigma ) \rightarrow \int _{\mathbb {R}^\gamma } \sigma _x \mu ^{+a} (d\sigma ), \qquad n\rightarrow +\infty . \end{aligned}$$

Since the sequence $\{\Lambda _n\}_{n\in \mathbb {N}}$ is exhausting, it contains a cofinal subsequence, $\{\Lambda _{n_k}\}_{k\in \mathbb {N}}$, such that also

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _x \Pi ^{\gamma ,\mathrm{Ising}}_{\Lambda _{n_k}} (d \sigma ) \rightarrow \int _{\mathbb {R}^\gamma } \sigma _x \nu ^{+} (d\sigma ), \qquad n\rightarrow +\infty , \end{aligned}$$

where $\Pi ^{\gamma ,\mathrm{Ising}}_{\Lambda _{n_k}}$ is the kernel (17) corresponding to the Ising single-spin measure (23), interaction intensities $a^2 \phi (|x-y|)$, and the choice $\bar{\sigma }_x = +1$ for all $x\in \gamma$. Thus, the validity of (28) will follow if we prove that, for each $\Lambda$ which contains $x$, the following holds

$$\begin{aligned} \int _{\mathbb {R}^\gamma } \sigma _x \Pi _{\Lambda }^{\gamma ,+a} (d \sigma ) \ge a \int _{\mathbb {R}^\gamma } \sigma _x \Pi ^{\gamma ,\mathrm{Ising}}_{\Lambda } (d \sigma ). \end{aligned}$$

(42)

Let $Z^\gamma _\Lambda (a)$ and $Z_\Lambda ^{^\gamma ,\mathrm{Ising}} (1)$ be the corresponding normalizing factors defined in (18). Then by (17) we have, cf. (15),

$$\begin{aligned}&\int _{\mathbb {R}^\gamma } \sigma _x \Pi _{\Lambda }^{\gamma ,+a} (d \sigma ) - a \int _{\mathbb {R}^\gamma } \sigma _x \Pi ^{\gamma ,\mathrm{Ising}}_{\Lambda } (d \sigma ) = \left( Z_\Lambda (a)Z_\Lambda ^{^\gamma ,\mathrm{Ising}} (1) \right) ^{-1} \nonumber \\&\quad \times \ \int _{\mathbb {R}^\gamma }\int _{\mathbb {R}^\gamma } (\sigma _x - a \tilde{\sigma }_x)\exp \left\{ \beta \sum _{\{x,y\}\subset \gamma _\Lambda }\phi (|y-z|) [\sigma _y \sigma _z + a^2\tilde{\sigma }_y \tilde{\sigma }_z]\right. \nonumber \\&\quad \left. + \sum _{y \in \gamma _\Lambda }[\sigma _y + a\tilde{\sigma }_y ]K_y \right\} \bigotimes _{x\in \gamma _\Lambda } \left( \chi ( d \sigma _x)\otimes \chi ^\mathrm{Ising} (d \tilde{\sigma }_x)\right) , \end{aligned}$$

(43)

where $\chi ^\mathrm{Ising}$ is given in (23) and $K_y = \beta a \sum _{z\in \gamma _{\Lambda ^c}: z \sim y } \phi (|y-z|)\ge 0$. Then (42) will follow from the positivity of the integral on the right-hand side of (43). Now we rewrite the integrand in (43) in the variables $u^{\pm }_x := (\sigma _x \pm a \tilde{\sigma }_x)/\sqrt{2}$, and then expand the exponent and write the integral as the sum of the products over $x\in \gamma _\Lambda$ of ‘one-site’ integrals having the form

$$\begin{aligned}&C_x \int _{\mathbb {R}^2} (u^+_x)^{m_x} (u^-_x)^{n_x} \chi ( d \sigma _x)\otimes \chi ^\mathrm{Ising} (d \tilde{\sigma }_x)\nonumber \\&\quad = C_x \int _{\mathbb {R}}\left[ (\sigma _x + a)^{m_x} (\sigma _x - a)^{n_x} + (\sigma _x - a)^{m_x} (\sigma _x + a)^{n_x} \right] \chi ( d\sigma _x), \quad C_x \ge 0.\qquad \quad \end{aligned}$$

(44)

Thus, to prove the statement we have to show that the integral on the right-hand side of (44) is nonnegative for all values of $m_x, n_x \in \mathbb {N}_0$. By the assumed symmetry of $\chi$, this integral vanishes if $m_x$ and $n_x$ are of different parity. If both $m_x$ and $n_x$ are even, then the positivity is immediate. Thus, it is left to consider the case where $m_x = 2 k+1$ and $n_x = 2l+1$. By the symmetry of $\chi$, it is enough to take $k\ge l$. Thus, we have to prove the positivity of the following integral

$$\begin{aligned}&\int _{\mathbb {R}}\left[ (\sigma + a)^{2k+1} (\sigma - a)^{2l+1} + (\sigma - a)^{2k+1} (\sigma + a)^{2l +1} \right] \chi ( d\sigma ) \\&\quad = 2 \int \limits _{0}^{+\infty }(\sigma ^2 - a^2)^{2l+1}\left[ (\sigma + a)^{k-l} + (\sigma + a)^{k-l}\right] \chi (d \sigma ). \end{aligned}$$

The function $\varphi (\sigma ) := (\sigma + a)^{k-l} + (\sigma + a)^{k-l}$ is increasing on $[ 0, +\infty )$. The integral on the right-hand side of the latter equality can be written in the form

$$\begin{aligned}&\int \limits _{0}^{+\infty }(\sigma ^2 - a^2)^{2l+1} \varphi (\sigma ) \chi ( d\sigma ) = I_1 (a) + I_2 (a) + I_3 (a),\nonumber \\&I_1 (a) := \int \limits _{0}^a (\sigma ^2 - a^2)^{2l+1} \varphi (\sigma ) \chi ( d\sigma ) \ge - a^{4l +2}\varphi (a) \chi ([0,a]), \nonumber \\&I_2 (a) := \int \limits _{a}^{a\sqrt{2}} (\sigma ^2 - a^2)^{2l+1} \varphi (\sigma ) \chi ( d\sigma ) \ge 0, \nonumber \\&I_3 (a) := \int \limits _{a\sqrt{2} }^{+\infty } (\sigma ^2 - a^2)^{2l+1} \varphi (\sigma ) \chi ( d\sigma ) \ge a^{4l +2}\varphi (a\sqrt{2}) \chi ([a\sqrt{2}, +\infty )) \end{aligned}$$

(45)

In view of (27), the sum on the right-hand side of (45) is nonnegative, which completes the proof.