A Phase Transition in a Quenched Amorphous Ferromagnet

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over $\mathbb{R}^d$, $d\geq 2$. 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 particle density (i.e., 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 R d , d ≥ 2, in such a way that every bounded Λ ⊂ 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 ∈ R d and spin σ x ∈ R. The locations of 'particles' constitute a locally finite set (configuration) γ ⊂ R d , and the spins take real values. We also assume that each σ x is characterized by one and the same symmetric a priori distribution χ on R. The interaction is supposed to be pair-wise and attractive. For the 'particles' located at x and y, the interaction energy is φ(|x − y|)σ x σ y , with φ(r) ≥ φ * > 0 for r ∈ [0, r * ], and φ(r) = 0 for r > r * . If γ 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 [28], further continued in [5,36]. In [27], a similar model living on a more general discrete metric space was studied. The next step was made in [25] 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 [25] 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 γ be random, obeying the Poisson law with homogeneous density λ > 0. The graph structure on γ is then defined by the spin-spin interaction: x, y ∈ γ are adjacent (neighbors) if φ(|x − y|) > 0. We call this model the amorphous ferromagnet, cf. [33,Section 11].
In view of the mentioned randomness, there are 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) configurationsγ = {(x, σ x ) : x ∈ γ}. 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 γ. These are quenched states, cf. [7]. The global observables characterizing such states ought to be self-averaging -taking the same value for all typical (i.e., for almost all) configurations γ. 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 γ, of thermodynamic states with properties suitable for physical applications. • Measurability of thermodynamic states with respect to γ.
• Multiplicity of such states, for almost γ, for temperatures T < T * , where the self-averaging parameter T * > 0 may depend on the model parameters λ, φ * , 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, cf. [11,19,20,21,35] 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 σ x = ±1, there exist methods applicable to the corresponding models on graphs, cf. [22,29]. For such models, see also [10], 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 γ and the adjacency relation x ∼ y defined by the property |x − y| ≤ r * . That is, the set of edges of the graph is ε γ = {{x, y} ⊂ γ : x ∼ y}, and the graph itself -known as the Gilbert graph -is then the pair (γ, ε γ ). It has various applications and is intensively studied, see, e.g., [4,17,30,32]. The probability distribution of (γ, ε γ ) is described in subsection 2.1 below. If this graph has an infinite connected component, which is a random event with probability dependent on λ, under certain conditions one can observe the Bernoulli bond percolation with a nontrivial percolation threshold q * ∈ (0, 1). We combine the mentioned methods and prove that the mean magnetization in the model can be positive for almost all configurations γ, and hence the quenched Gibbs states can be multiple, if the particle density λ and the inverse temperature β = 1/T are large enough 1 . Finally, let us mention that, for our model with σ x ∈ R, the problem of uniqueness of Gibbs states remains open, see subsection 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 π λ we denote the Poisson measure with density λ > 0 which describes the probability distribution of the configurations γ. In Proposition 2.4 and Theorem 2.7 below, we show that there exists a set of configurations A 1 such that: (a) π λ (A 1 ) = 1 for all λ > 0; (b) G t (β|γ) = ∅ for all β > 0 and all γ ∈ A 1 . Here G t (β|γ) is the set of tempered Gibbs measures of our spin system on γ at a given β. Tempered Gibbs measures are Gibbs measures supported on the configurations with tempered growth of |σ x | as |x| → +∞. 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 |σ x |, whereas for typical ferromagnetic configurations in physical substances, most of the spins take values close to same s > 0. The proof of Theorem 2.7 is based on [25, Theorem 3.1] and on the property of π λ obtained in Proposition 2.4, in which A 1 appears as the set of all those configurations for which the quantities in (2.9) are finite, and hence the graph (γ, ε γ ) belongs to the class of sparse graphs introduced in [25]. The Ising model on γ is a particular case of our model, which corresponds to the choice χ(dσ) = δ(σ 2 − 1)dσ. As is well-known, the set of the Gibbs measures of this model, G Ising t (β|γ), is nonempty for all γ. Next, for by the Wells correlation inequality [40], it follows that, cf. Proposition 3.3 below, where a > 0 is determined by the measure χ, see (3.3). For the reader convenience, we present here a complete proof of the Wells inequality, which is a refinement of that in [9,Appendix]. By standard results on the continuum percolation driven by the Poisson random point process, see [30,32] and also [16,Corollary 3.7] and [17,Theorem 3.1], it follows that, for π λ -almost all γ, the corresponding graph (γ, ε γ ) has an infinite connected component whenever λ > λ * , where a non-random parameter λ * > 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 (γ, ε γ ). As in the previous result, it is possible to show, see Propositions 2.2 and 2.3 below, that for λ > λ * there exists q * ∈ (0, 1) such that (1.2) π λ (A 3 (q)) = 1, for q > q * and λ > λ * , where A 3 (q) is the set of γ such that the Bernoulli bond percolation with bond probability q occurs on the infinite connected component of (γ, ε γ ). By [22, Theorem 2.1], we know that 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 [17,30,32] or a tempered Gibbs random field, see [16,Corollary 3.7]. The only condition is that the graph almost surely has the summability property as in Proposition 2.4, see [13] for more detail.
where |A| stands for the cardinality of A. The space Γ is equipped with the vague topology being the weakest one in which the maps Γ ∋ γ → x∈γ f (x) are continuous for all continuous functions f : R d → R with compact support, see, e.g., [3]. By B(Γ) we denote the corresponding Borel σ-field. The vague topology is metrizable in such a way that the corresponding metric space is complete and separable. For λ > 0, by π λ we denote the homogeneous Poisson measure on (Γ, B(Γ)) with intensity (density) λ. Note that the set of all finite configurations Γ 0 is a Borel subset of Γ such that π λ (Γ 0 ) = 0. That is, π λ -almost all configurations γ ∈ Γ are infinite.
Each γ can be considered as a graph with vertex set γ and the adjacency relation x ∼ y defined by the condition |x − y| ≤ r * . Then ε γ = {{x, y} ⊂ γ : x ∼ y} is its edge set. The probability distribution of the random graph (γ, ε γ ) is constructed in the following way, cf. [16]. Let ε denote a set of pairs of distinct points, i.e., of e = {x, y}, x, y ∈ R d , x = y. We say that ε is locally finite if ε Λ is finite for each Λ ∈ B b (R d ). Here ε Λ := {{x, y} ∈ ε : {x, y} ⊂ Λ}. Let E be the set of all locally finite ε, and F(E) be the σ-field of subsets of E generated by the counting maps Note that E is, in fact, the set of locally finite configurations, cf. (2.1), over the symmetrization of the set Let P(E) be the set of all probability measures on (E, F(E)). Each ς ∈ P(E) can uniquely be determined by its Laplace transform, which we introduce in the following way. Let θ : R d ×R d → (−1, 0] be measurable, symmetric, and local, i.e., θ(x, y) = 0 whenever x or y is in Λ c := R \ Λ for some Λ ∈ B b (R d ). By Θ we denote the set of all such functions. Then for each ς ∈ P(E), its Laplace transform is defined as be measurable and symmetric. For each θ ∈ Θ, the pointwise product gθ is also in Θ. An independent g-thinning of a given ς ∈ P(E), cf. [14, Section 11.2], is the measure ς g defined by the relation The g-thinning of ς means that each configuration ε distributed according to ς is 'thinned' in the sense that each {x, y} ∈ ε is removed from the edge configuration with probability 1−g(x, y) and is kept with probability g(x, y). The probability distribution of such 'thinned' configurations is then ς g . Given γ ∈ Γ, the measure ς(·|γ) ∈ P(E) is defined by its Laplace transform where j(x, y) = 1 for |x−y| ≤ r * and j(x, y) = 0 otherwise. For a measurable Ψ : Lemma 2.4]. The probability distribution of the graph (γ, ε γ ) is now defined by the measure It may happen that typical graphs (γ, ε γ ) have only finite connected components. Clearly, the probability of this event, calculated from (2.4), depends on λ. Given g as in (2.2), by ς g (·|γ) we denote the independent g-thinning of ς(·|γ), and, cf. (2.4), Then by Proposition 2.1 we obtain the following fact.
Indeed, by taking in (2.5) g(x, y) = j(x, y), we prove Proposition 2.2 with For a constant function q(x, y) ≡ q ∈ [0, 1], let us consider the independent q-thinning of ς(·|γ). 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 (γ, ε γ ). The next fact can also be deduced from Proposition 2.1. Proposition 2.3. Let λ * be as in Proposition 2.2 and inequality λ > λ * hold. Then, there exists q * ∈ (0, 1) such that for π λ -almost all γ, the Bernoulli bond percolation with bond probability q > q * occurs on the infinite connected component of (γ, ε γ ).
By the definition of the Poisson measure π λ , for each n ∈ N and any measurable and symmetric function f : Next, we rewrite (2.9) in the form Applying this in (2.11) we get By the latter and (2.10), as well as by the translation invariance of π λ , we then obtain from (2.11) Here and V (2r * ) = R d I(x)dx is the volume of the ball B 2r * (0). The latter estimate leads to the conclusion π λ ({γ : a γ (α, θ) = +∞}) = 0, 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 χ on R. We assume that, for some u > 2 and κ > 0, the following holds For a fixed γ, let R γ stand for the space of all maps σ : γ → R. We equip it with the topology of point-wise convergence and the corresponding Borel σ-field B(R γ ). Let also P(R γ ) denote the set of all probability measures on (R γ , B(R γ )). For Λ ∈ B(R d ) and γ ∈ Γ, we set γ Λ = γ ∩ Λ and denote by σ Λ the restriction of σ to γ Λ , i.e., σ Λ = (σ x ) x∈γ Λ . For σ,σ ∈ R γ and Λ ∈ B(R d ), by σ Λ ×σ Λ c we denote the element σ ′ ∈ R γ such that σ ′ Λ = σ Λ and σ ′ Λ c =σ Λ c . The ferromagnet that we study is characterized by a ferromagnetic spinspin interaction, which for fixed Λ ∈ B b (R d ) and γ ∈ Γ is described by the following relative energy functional Here φ : R + → R + is a measurable and bounded function such that, φ(r) ≥ φ * > 0 for r ∈ [0, r * ] and φ(r) = 0 for all r > r * .
is the corresponding Borel σ-field. The algebra of local sets is We will use the following topology on P(R γ ), see [15, Definition 4.2, page 59].
Definition 2.5. The topology of local set convergence (L-topology for short) is the weakest topology on P(R γ ) that makes the evaluation maps µ → µ(A) continuous for all A ∈ B loc (R γ ).
where I A is the indicator of A ∈ B(R γ ), E γ Λ is as in (2.13), and Thus, for each A ∈ B(R γ ), Π γ Λ (A|·) is B(R γ )-measurable, and, for each σ ∈ R γ , Π γ Λ (·|σ) is a probability measure on (R γ , B(R γ )). The collection of probability kernels {Π γ Λ : Λ ∈ B b (R d )} is called the Gibbs specification of the model we consider, see [15,Chapter 2]. It enjoys the consistency property which holds for all A ∈ B(R γ ),σ ∈ R γ , and all Λ 1 , Definition 2.6. A probability measure µ on (R γ , B(R γ )) is said to be a quenched Gibbs measure (for a fixed γ) if it satisfies the Dobrushin-Lanford-Ruelle equation The set of all such measures is denoted by G(β|γ).
In modern equilibrium statistical mechanics, the notion of thermodynamic phase of a system of bounded spins living on a fixed graph like Z d is attributed to the extreme elements of the set of corresponding Gibbs measures, see, e.g., [15,Chapter 7] or [38,Chapter 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 [28,5,36] or a more recent development in [2,27] and [1,Chapter 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 measures 2 . We take this approach and study quenched Gibbs measures introduced in Definition 2.6 with a priori prescribed support properties. We call them tempered Gibbs states. Thus, for an α > 0, we define where the weights w α are as in (2.8). For each fixed γ, Σ(α) is a Borel subset of R γ and its elements are called tempered configurations. Then is the set of tempered Gibbs states.
Theorem 2.7. Let the single-spin measure χ be such that (2.12) holds and A 1 be as in Proposition 2.4. Then, for all γ ∈ A 1 and all β > 0, the set of Gibbs states G t (β|γ) is nonempty. Moreover, for each γ ∈ A 1 , and for each positive ϑ and α, there exists a finite C γ (ϑ, α) > 0 such that the estimate holds uniformly for all µ ∈ G t (β|γ).
Proof. The proof of all the statements of the theorem follows by Theorem 1 of [25] since all the conditions of that theorem are satisfied in view of Proposition 2.4 and the assumed properties of χ.
Let us make some comments. The existence of Gibbs measures follows from the relative compactness of the family {Π γ Λ (·|σ) : Λ ∈ B b (R d )} in the L-topology, see Definition 2.5, for at least someσ ∈ Σ(α). A typical choice ofσ, for which the compactness is proven, isσ x = s ∈ R for all x ∈ γ. Note that suchσ is tempered, see (2.17). Then the accumulation points of the family {Π γ Λ (·|σ) : Λ ∈ B b (R d )} are shown to obey the Dobrushin-Lanford-Ruelle equation and to satisfy the estimate in (2.19), in which the constant C γ (ϑ, α) can be expressed explicitly in terms of the weights as in (2.8) and the parameters defined in (2.9), cf. Proposition 2.4 above. Thus, the accumulation points are tempered measures, and hence belong to G t (β|γ). Note that G t (β|γ) is nonempty for all those γ, for which both a γ (α, θ) and b γ (α) are finite. By similar arguments, one can show that 2 Note that the theory of quantum stabilization and phase transitions in quantum anharmonic crystals developed in [2,24,27] and [1,Chapter 6] with the use of tempered Gibbs measures is consistent with the corresponding phenomena observed experimentally. G t (β|γ) is compact in the L-topology. Finally, let us mention that u > 2 in (2.12) and θ in Proposition 2.4 should be such that θ(u − 2) > 2. Under this condition the sufficiently fast decay of the tail of χ compensates destabilizing effect of the property (2.7) of the underlying graph, see [25] for more detail. Since Proposition 2.4 holds for all θ > 0, then we just assume that u > 2. A , n ∈ N, and each Λ ∈ B b (R d ) is contained in a certain Λ n . Givenσ ∈ Σ(α), the relative compactness of the family {Π γ Λ (·|σ) : Λ ∈ B b (R d )} yields that there exists a cofinal sequence {Λ n } n∈N such that the sequence {Π γ Λn (·|σ)} n∈N converges in the L-topology to a certain element of G t (β|γ). For a > 0, by (2.20) µ ±a ∈ G t (β|γ) we denote limiting elements of G t (β|γ) which correspond toσ x = ±a for all x ∈ γ. Each such µ ±a depends on the sequence {Λ n } n∈N along which it has been attained. Note that only limiting Gibbs measures can approximate thermodynamic states of large finite systems, see [15, Section 7.1]. Now we turn to the single-spin measure χ. If it has compact support, as was the case in [35], then (2.12) clearly holds for any u and κ. The most known example of such χ is which corresponds to an Ising magnet. Here δ s is the Dirac measure concentrated at s ∈ R. We reserve a special notation G Ising (β|γ) for the set of all corresponding Gibbs measures. By ν ± ∈ G Ising (β|γ), we denote the limiting Gibbs measures as in (2.20) with a = 1. In this case, however, ν ± are independent of the sequences {Λ n } n∈N along which they were attained. This holds because, for each x and ν ∈ G Ising (β|γ), That is, ν + and ν − are the maximum and minimum elements of G Ising (β|γ), respectively, cf. [27,Theorem 3.8].
In the case of 'unbounded' spins, a natural choice of the single-spin measure is where V : R → R is a measurable even function such that: (a) the set {t ∈ R : V (t) < +∞} is of positive Lebesgue measure; (b) V (t) increases at infinity as |t| u+ǫ with some ǫ > 0 and u being as in (2.12). This includes the case where V is a polynomial of even degree at least 4 with positive leading coefficient, cf. [25,26,27,28].

The question of uniqueness.
Once the existence of Gibbs states has been established, the next the problem of their uniqueness/nonuniqueness arises. Then a phase transition is understood as the possibility to pass from the uniqueness to nonuniqueness by changing the relevant parameters of the model. Thus, prior to proving non-uniqueness of µ ∈ G t (β|γ), which holds for π λ -almost all γ whenever β and λ are large enough, see Theorem 3.1 below, we address the question of whether the same uniqueness does actually hold for some values of these parameters. For small enough λ, all the connected components of the graph (γ, ε γ ) are finite, and hence G t (β|γ) is a singleton for all β. On the other hand, G t (β|γ) is a singleton if and only if, for each x ∈ γ, arbitraryσ ∈ Σ(α), and any cofinal sequence {Λ n } n∈N , one has This equivalence holds for any symmetric ferromagnet satisfying the bound in (2.19), that can be proven by standard arguments based on the Strassen theorem, see [27] for more detail. Actually, for models with 'unbounded' spins living on an infinite connected graph with globally unbounded degree, which by (2.7) is the case in our situation, there are no tools 3 for proving (2.22). For the Ising ferromagnet, the uniqueness in question can be obtained by percolation arguments, see [18,Theorem 7.2].

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 G t (β|γ) for π λ -almost all γ contains at least two elements if β and λ are big enough. It is equivalent to the appearance of a nonzero magnetization in states µ ±a ∈ G t (β|γ), cf. [15,Chapter 19] and (2.22). 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 (γ, ε γ ) possess an infinite connected component, which holds for π λ -almost all γ whenever λ > λ * , see [30,32] and also [16,Corollary 3.7] and [17,Theorem 3.1]. For λ < λ * , we have no infinite connected component of (γ, ε γ ) and thus |G(β|γ)| = 1 for all β and π λ -almost all γ. 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 (γ, ε γ ), established in Propositions 2.2 and 2.3, and the existence of multiple Gibbs states in the corresponding Ising model, established in [22]. Our goal is to prove the following result.
Theorem 3.1. Let the measure χ be as in Theorem 2.7 and such that χ({0}) < χ(R). Assume also that the intensity λ of the underlying Poisson point process satisfies the condition λ > λ * , and thus the typical graph (γ, ε γ ) has an infinite connected component. Then there exists a constant β * > 0 such that, for β > β * and π λ -almost all γ, the sets G t (β|γ) contain at least two elements.
The proof of this statement is based on the following result, cf. (2.20).
Lemma 3.2. Let the conditions of Theorem 3.1 be satisfied. Then there exist a > 0, and β * > 0 such that, for all β > a 2 β * , all µ +a ∈ G(β|γ), and some o ∈ γ, the following estimate holds for π λ -almost all γ: 3 The celebrated Dobrushin uniqueness technique is not applicable here.
The proof of this lemma is given in the next subsection.
Proof of Theorem 3.1: Since the integral in (3.1) is the limit of those in (2.22) withσ x = a, then (3.1) contradicts (2.22) and hence implies nonuniqueness, which ought to hold for β > β * := a 2 β * . On the other hand, by the invariance of χ and of the interaction in (2.13) with respect to the transformation σ → −σ andσ → −σ, we have Then (3.1) yields µ +a = µ −a and hence the multiplicity in question. Note that o in (3.1) belongs to the infinite connected component of (γ, ε γ ), and the integral in (3.1) is the mean value of the spin at this vertex in state µ +a .

Proof of Lemma 3.2.
First, by means of the percolation arguments of [22], 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 (2.21), G Ising (β|γ) denotes the set of all corresponding Gibbs measures, and ν + ∈ G Ising (β|γ) is the maximum Gibbs measure as in (2.20) with a = 1. We are going to use the key fact proven in [22]: the Ising model with constant intensities φ(|x − y|) = φ * > 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 * ∈ (0, 1) if β > [log(1 + q * ) − log(1 − q * )]/2φ * . In our case, for π λ -almost all γ, the graph γ admits this percolation and the threshold probability satisfies q * ≥ λ * /λ, see Proposition 2.3. Then, for some o ∈ γ, it follows that which together with (3.2) yields the proof in this case. Now we turn to the general case and estimate the integral in (3.1) from below by the corresponding integral with respect to the maximum Gibbs measure ν + of the Ising model with a rescaled interaction intensity. The proof of the lemma immediately follows from the Wells inequality used, e.g., in [34].  Then, for each x ∈ γ and each µ +a ∈ G(β|γ), as well as for ν + ∈ G Ising (a 2 β|γ), we have that

14ALEXEI DALETSKII, YURI KONDRATIEV, YURI KOZITSKY, AND TANJA PASUREK
As the original publication [40] is hardly attainable and the proof in [9] contains numerical inaccuracies, for the reader convenience in Section 5 we give a short proof of this inequality in the form suitable for our purposes.

Random Gibbs measures
Random Gibbs measures of spin systems are supposed to depend on the random parameters (γ in ours case) in a measurable way, see, e.g., [8, Definition 6.2.5] and [26,Definition 2.3]. At the same time, the measures introduced in Definition 2.6 and (2.18) are defined on spaces which themselves depend on γ, and thus one cannot speak of the corresponding measurability in this setting. In order to settle this problem, we use the results of [13] 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 4.1 below.
For our model, the space of marked configurations is where Γ is as in (2.1). Thus, the elements of Γ consist of pairs (x, σ x ), and γ =γ ′ implies γ = γ ′ . In the sequel, we use the canonical projection and equip Γ with the following topology. Let f : R d × R → R be a bounded continuous function with support contained in Λ × R for some Λ ∈ B b (R d ).
The topology in question is the weakest one which makes the maps continuous for all possible f : R d × R → R as above. This topology is completely and separably metrizable, see [12,Section 2], and thus Γ is a Polish space. Let B( Γ) be the corresponding Borel σ-field. From the definition of the topologies of Γ and Γ, it follows that the projection defined in (4.1) is continuous, and hence, for each γ ∈ Γ, we have that For each fixed γ ∈ Γ, R γ is a Polish space embedded into Γ, which is a Polish space as well. By the Kuratowski theorem [37, page 21], the latter implies that the Borel σ-fields B(R γ ) and are measurably isomorphic. Thus, any probability measure µ on B( Γ) with the property µ(R γ ) = 1 can be redefined as a measure on B(R γ ). Let P( Γ) be the set of all probability measures on ( Γ, B( Γ)). We equip it with the topology defined as follows. For a fixed Λ ∈ B b (R d ), let B Λ ( Γ) be the smallest σ-subfield of B( Γ) such that the maps as in (4.2) are B Λ ( Γ)measurable for all bounded measurable functions f : R d × R → R with support contained in Λ × R. Then we set, cf. (2.14), Now the L-topology on P( Γ) is defined as in Definition 2.5 by using B loc ( Γ) as the algebra of local sets. A map Γ ∋γ → ϕ(γ) ∈ R is called local if it is B Λ ( Γ)-measurable for some Λ ∈ B b (R d ). Local maps Γ ∋ γ → ϕ(γ) ∈ R are defined in the same way.
Letσ in (2.15) be fixed in such a way that σ x = s for some s ∈ R and all x ∈ γ. Then, for each γ ∈ Γ and Λ ∈ B b (R d ) and for suchσ, we can define a probability measure on ( Γ, B( Γ)) by setting is given in (2.15). By the very construction, the map Γ ∋ γ → Π s Λ (A|γ) ∈ R is measurable for each A ∈ B( Γ). Thus, we can consider which is an element of P( Γ), equipped with the L-topology defined above. It can be shown, see [13,Corollary 4.2], that for our model the family Thus,η s Γ = π λ , and we can disintegrate, cf. (4.4), and obtain where η s is a regular conditional measure such that η s (A|γ) = η s (A ∩ R γ |γ) for almost all γ. As in (4.3), we then redefine η s (·|γ) as a measure on R γ , for which we keep the same notation. One can prove [13] that, for almost all γ, η s (·|γ) ∈ G t (β|γ). Thus, η s (·|γ) is a random Gibbs measure.
In principle, we could construct such Gibbs measures without the study performed in Section 2, just by showing that the family {η s Λ } Λ∈B b (R d ) is relatively compact in P( Γ). However, this way has the following drawbacks: (a) in contrast to those in (2.20), the measures η s (·|γ) need not be limiting and hence cannot approximate thermodynamic states of large finite systems; (b) there is no control on the sets of γ, as well as on their dependence on λ and s, for which η s (·|γ) exist, cf. Proposition 2.4; (c) nothing can be said of the integrability properties of η s (·|γ), cf. (2.19); (d) it is unclear whether we can have η +a = η −a . These problems are partly resolved in the statement below. Recall that each µ +a is a measure on R γ , γ ∈ A 1 , see Proposition 2.4, and is attained along a cofinal sequence L := {Λ n } n∈N , that will be indicated as µ +a L . For each γ ∈ A 1 , such measures µ +a constitute the set of accumulation points of the family {Π γ Λ (·|σ) : Λ ∈ B b (R d )}. The meaning of the theorem below is that, for a full π λ -measure subset A ′ 1 ⊂ A 1 , there exists a (measurable) selection {µ +a L(γ) } γ∈A ′ 1 such that µ +a L(γ) = η +a (·|γ) for all γ ∈ A ′ 1 .
Proof. From now on we fix s = +a and {Λ n } n∈N , and henceη s . Note that we cannot expect that the assumed convergenceη s Λn →η s does imply that Π γ Λn (·|σ) → η s (·|γ), which would yield the proof.
Fix dense subsets {y n k } k∈N ⊂ Ξ n . As in the proof of Lemma 2.3 in [23, page 20], by means of this partition we introduce a linear order of the elements of each γ ∈ Γ. If x, y ∈ γ belong to distinct Ξ n , we set x < y whenever x ∈ Ξ n , y ∈ Ξ m , and n < m. If both x and y lie in the same Ξ n , let k be the smallest integer such that |x − y n k | = |y − y n k |. Then we set x < y if |x − y n k | < |y − y n k |. Next, we enumerate the elements of γ in accordance with the order in such a way that x 1 < x 2 < · · · < x k < · · · . This defines an enumeration on Γ, that is, the map such that all (x k , σ x k ) are distinct and (x k , σ x k ) ∈γ. This map is measurable in the sense that {γ : x k ∈ ∆, σ x k ∈ Σ} ∈ B( Γ) for each k ∈ N, ∆ ∈ B b (R d ) and Σ ∈ B(R). The latter fact can be proven by a slight generalization of the proof of Lemma 2.3 in [23, page 20]. Then, for a fixed ∆ ∈ B b (R d ) and any k ∈ N, p ∈ (Q d ) N , and q ∈ Q N , the real and imaginary parts of the functionγ → exp[i(p k · x k + q k σ x k )] ∈ C are as that in (4.2), and hence are B ∆ ( Γ)-measurable. Here i = √ −1 and p · x stands for the scalar product in R d . Let D = {∆ n } n∈N ⊂ B b (R d ) be a cofinal sequence. For ǫ as in (4.6), ∆ ∈ D, p ∈ (Q d ) N , and q ∈ Q N , we set where γ =p(γ), (x k , σ x k ) is the k-th element of the sequence ǫ(γ), and the sum runs over the set of all those j for which x k j ∈ ∆. By construction, each such f is B ∆ ( Γ)-measurable. Let 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 Γ. The latter means that, for any two distinctγ,γ ′ ∈ Γ, one finds f ∈ F such that f (γ) = f (γ ′ Since h ∆,q ∈ L 1 (Γ, dπ λ ), we can integrate and by (4.5), (4.8), (4.9), and (4.10) obtain which holds for all ∆ ∈ D, p ∈ (Q d ) N , and q ∈ Q N . This yields the proof since the integrand functions constitute separating classes.

Appendix: Proof of Proposition 3.3
For the general choice of χ, let Π γ,+a Λ be defined as in (2.15) withσ x = +a for all x ∈ γ. Each µ +a is the weak limit of {Π γ,+a Λn } n∈N for some cofinal sequence {Λ n } n∈N . In the case of unbounded spins, this convergence alone does not yet imply the convergence of the moments of Π γ,+a Λn to that on the left-hand side of (3.4). Then we use the uniform in n bound as in (2.19), which can also be proven for all Π γ,+a Λ , and obtain R γ σ x Π γ,+a Λn (dσ) → R γ σ x µ +a (dσ), n → +∞.
Since the sequence {Λ n } n∈N is exhausting, it contains a cofinal subsequence, {Λ n k } k∈N , such that also where Π γ,Ising Λn k is the kernel (2.15) corresponding to the Ising single-spin measure (2.21), interaction intensities a 2 φ(|x − y|), and the choiceσ x = +1 for all x ∈ γ. Thus, the validity of (3.4) will follow if we prove that, for each Λ which contains x, the following holds where χ Ising is given in (2.21) and K y = βa z∈γ Λ c :z∼y φ(|y − z|) ≥ 0. Then (5.1) will follow from the positivity of the integral on the right-hand side of (5.2). Now we rewrite the integrand in (5.2) in the variables u ± x := (σ x ± aσ x )/ √ 2, and then expand the exponent and write the integral as the sum of the products over x ∈ γ Λ of 'one-site' integrals having the form Thus, to prove the statement we have to show that the integral on the righthand side of (5.3) is nonnegative for all values of m x , n x ∈ N 0 . By the assumed symmetry of χ, 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 = 2k + 1 and n x = 2l + 1. By the symmetry of χ, it is enough to take k ≥ l. Thus, we have to prove the positivity of the following integral R (σ + a) 2k+1 (σ − a) 2l+1 + (σ − a) 2k+1 (σ + a) 2l+1 χ(dσ) = 2 +∞ 0 (σ 2 − a 2 ) 2l+1 (σ + a) k−l + (σ + a) k−l χ(dσ).