Bassalygo-Dobrushin uniqueness for continuous spin systems on irregular graphs

The theory of Gibbs random fields indexed by countable sets (e.g., Gibbs states of lattice models) is now well elaborated, see [1]. At the same time, only a few papers deal with the field having index sets presented by irregular graphs. Among them there is a paper by L. A. Bassalygo and R. L. Dobrushin, [2]. It presents a technique of proving uniqueness if the single-site state spaces (single-spin spaces) are finite. Our aim is to develop a similar technique, which covers the case where the single-site state spaces are general metric spaces. In this case, with slight abuse of terminology we say that the spins are continuous, which is reflected in the title above. The basic assumption, however, is that the interaction between the spins is bounded. As possible applications of our theory we mention the theory of Euclidean Gibbs states of the following quantum models. In each of them, quantum particles are located at sites (one particle per site), which form an irregular structure. The single-particle Hamiltonians have discrete spectra and the interparticle interaction is pair-wise and bounded. As a quantum particle, one can take: (a) a free particle moving in a compact Riemannian manifold (e.g. quantum rotator), see [3,4]; (b) a free particle moving in a compact subset of R; (c) a quantum anharmonic oscillator, see e.g. [5]. Let (L,E) be a graph with (infinite) countable sets of vertices, L, and edges, E. We also suppose that the graph is simple, i.e., it has no loops, isolated vertices, and multiple edges. Two vertices `, ` are called adjacent if there exists an edge 〈`, `〉. The number n` of the vertices adjacent to ` is called degree. For each ` ∈ L, let X` be a complete separable metric space (Polish space), B(X`) be the corresponding Borel σ-field, and χ` be a finite Borel measure on (X`,B(X`)). For an edge 〈`, `〉, let a bounded symmetric continuous function (potential) V``′ : X` × X`′ → R be given. Under certain conditions, these objects define a Gibbs random field on the product space X = ∏ `∈LX`. If all V``′ equal zero, there exists only one Gibbs field. Thus, one can expect the same uniqueness if the potentials are sufficiently small, which certainly depends on the underlying graph. If the latter is regular (each vertex has the same degree), the proof of the uniqueness by small potentials is quite standard. The case where n`’s are different but globally bounded (there exists n̂ ∈ N, such that n` 6 n̂ for all ` ∈ L) can be handled similarly. The situation changes substantially if sup`∈L n` = +∞. This can be seen from the example considered in Section 4 below, where the graph is so dense that the ferromagnetic Ising model defined on this graph has multiple Gibbs states for arbitrary non-zero interactions. For sparse graphs of a certain kind, which in


Introduction and setup
The theory of Gibbs random fields indexed by countable sets (e.g., Gibbs states of lattice models) is now well elaborated, see [1].At the same time, only a few papers deal with the field having index sets presented by irregular graphs.Among them there is a paper by L. A. Bassalygo and R. L. Dobrushin, [2].It presents a technique of proving uniqueness if the single-site state spaces (single-spin spaces) are finite.Our aim is to develop a similar technique, which covers the case where the single-site state spaces are general metric spaces.In this case, with slight abuse of terminology we say that the spins are continuous, which is reflected in the title above.The basic assumption, however, is that the interaction between the spins is bounded.As possible applications of our theory we mention the theory of Euclidean Gibbs states of the following quantum models.In each of them, quantum particles are located at sites (one particle per site), which form an irregular structure.The single-particle Hamiltonians have discrete spectra and the interparticle interaction is pair-wise and bounded.As a quantum particle, one can take: (a) a free particle moving in a compact Riemannian manifold (e.g.quantum rotator), see [3,4]; (b) a free particle moving in a compact subset of R d ; (c) a quantum anharmonic oscillator, see e.g.[5].
Let (L, E) be a graph with (infinite) countable sets of vertices, L, and edges, E. We also suppose that the graph is simple, i.e., it has no loops, isolated vertices, and multiple edges.Two vertices , are called adjacent if there exists an edge , .The number n of the vertices adjacent to is called degree.For each ∈ L, let X be a complete separable metric space (Polish space), B(X ) be the corresponding Borel σ-field, and χ be a finite Borel measure on (X , B(X )).For an edge , , let a bounded symmetric continuous function (potential) V : X × X → R be given.Under certain conditions, these objects define a Gibbs random field on the product space X = ∈L X .If all V equal zero, there exists only one Gibbs field.Thus, one can expect the same uniqueness if the potentials are sufficiently small, which certainly depends on the underlying graph.If the latter is regular (each vertex has the same degree), the proof of the uniqueness by small potentials is quite standard.The case where n 's are different but globally bounded (there exists n ∈ N, such that n n for all ∈ L) can be handled similarly.The situation changes substantially if sup ∈L n = +∞.This can be seen from the example considered in Section 4 below, where the graph is so dense that the ferromagnetic Ising model defined on this graph has multiple Gibbs states for arbitrary non-zero interactions.For sparse graphs of a certain kind, which in particular means that the vertices of large degree are at large distances from each other, we prove that the number of Gibbs fields with Polish single-spin spaces is exactly one if the potentials are small enough.The proof is based on an extension and refinement of the technique developed in [2], where all X were finite.
For a finite set A, by |A| we denote its cardinality (the number of elements in A).Let L fin stand for the family of all finite non-void subsets of the vertex set L. A property related with a given Λ ∈ L fin is called local, whereas global properties are going to be related with the whole graph.If we say that something holds for all (resp.for all e), it holds for all ∈ L (resp.all e ∈ E).As usual, for Λ ⊂ L, we write Λ c = L \ Λ.A sequence D ⊂ L fin is called cofinal if it is ordered by inclusion and exhausts L. The latter means that any Λ ∈ L fin is contained in a certain ∆ ∈ D. The limits taken along such a sequence will be denoted by lim D .For Λ ⊂ L, we set The latter sets are called the edge and the vertex boundary of Λ, respectively.For a one-point Λ = { }, its edge and vertex boundaries are written ∂ E and ∂ L respectively.Clearly, the cardinalities of these both sets coincide, that is, We suppose that the graph is locally finite, which means that, for every , its degree In principle, a path may intersect itself, that is, certain vertices k may appear in ϑ( , ) more than once.The length of a path, |ϑ( , )|, is set to be the number of pairs of consecutive vertices; thus, in the situation above we had |ϑ( , )| = n.A path in L is merely called a path.By ρ( , ) we denote the distance between and -the length of the shortest path connecting these vertices.Definition 1.1.A path ϑ( , ) is called a self-avoiding if it has no self-intersections.This means that each i appears in ϑ( , ) only once.Thus, in a self-avoiding path, only the endpoints 0 , n can have degree 1.
For a vertex and N ∈ N, let L N ( ) be the set of all self-avoiding paths of length N originating at .
Definition 1.2.The graph is called sparse if there exist positive constants C and η such that, for every vertex , there exists N ∈ N such that, for all N N , (1.2) A particular case of locally bounded graphs was introduced in [2].For each of them, where φ : N → [1, +∞) is an increasing function, specific for the graph, such that By (1.3), two vertices of large degrees repel each other.Below, see Lemma 3.2, we prove that such graphs are sparse in the sense of Definition 1.2.
For a topological space Y , by C b (Y ) we denote the set of all bounded continuous functions f : Y → R, and by B(Y ) -the corresponding Borel σ-field.By saying that µ is a measure on Y , we mean that µ is a measure on the measurable space (Y, B(Y )).The set of all probability measures on Y is denoted by P(Y ).For a measurable function f : Y → R, we write Let X , ∈ L, be the Polish spaces mentioned above.For ∆ ⊆ L, the Cartesian product is equipped with the product topology, so that (X ∆ , B(X ∆ )) becomes a standard Borel space.This means that there exists a Polish space Y and a measurable isomorphism ϕ : X ∆ → Y .For more details we refer the reader to section 4.A, page 73 of [1].The elements of X ∆ are denoted by x ∆ ; we write X = X L and x = x L .For Λ ⊂ ∆, the just a position , where x 0 is a certain fixed element of X.In view of this embedding, one has B(X Λ ) ⊂ B(X ∆ ); thus, one can consider which is called the σ-field of local events.The tail σ-field is defined to be Suppose now that we have given finite Borel measures χ , ∈ L, and symmetric bounded continuous functions As usual, we write For Λ ∈ L fin , we set where and Clearly, each such π Λ is a probability kernel on (X, B(X)).This means that π Λ (•|y) ∈ P(X) for any y ∈ X, and π Λ (B|•) is measurable for any B ∈ B(X).The family {π Λ } Λ∈L fin defines a Gibbs random field on the graph in the following sense.By construction, which holds for any Λ ⊂ ∆.
The set of all such Gibbs random fields will be denoted by G.If necessary, we write G(V ) to indicate the dependence on the choice of V = (V ) , ∈E .The following property, the proof of which is quite standard, gives an important information about G. Proposition 1.4 (Feller Property).For every Given y ∈ X, let P y be the family of the accumulation points of {π Λ (•|y)} Λ∈L fin endowed with the usual weak topology defined by C b (X).
Corollary 1.5.For any y ∈ X, it follows that P y ⊂ G.
Proof.As C b (X) is a measure defining class, the inclusion in question can be obtained by showing that, for every µ ∈ P y and Λ ∈ L fin , (1.14)By supposition, there exists a cofinal sequence Let Λ be as in (1.14).Then one finds ∆ 0 ∈ D such that Λ ⊂ ∆ for all ∆ ∈ D starting from this ∆ 0 .For such ∆, one has (1.15) Passing here to the limit along D and taking into account that π This statement allows for the following generalization.By Definition 1.3, a convex combination of Gibbs fields is again a Gibbs field.Let G ex be the set of all extreme elements of G, i.e., those which cannot be presented as nontrivial convex combinations of other elements of G.If G = ∅, then G ex = ∅ and each Gibbs field can uniquely be represented as a convex combination of the elements of G ex , see Theorem 7.26, page 133 in [1].In particular, |G| = 1 if |G ex | = 1.Given µ ∈ G is extreme if and only if it is trivial on the tail σ-field (1.7), see Theorem 7.7, page 118 in [1].This means that µ(A) = 1 or µ(A) = 0 for any A ∈ B tail .Proposition 1.6.For every µ ∈ G ex and any cofinal sequence D, the sequence {π ∆ (•|y)} ∆∈D converges weakly to µ for all y ∈ A, where A ∈ B tail may depend on D and is such that µ(A) = 1.
Proof.For any f ∈ C b (X), one has lim D π ∆ (f |y) = µ(f ) for all y ∈ A f , such that A f ∈ B tail and µ(A f ) = 1, see claim (a) of Theorem 7.12, page 122 in [1].But the weak topology is metrizable.Therefore, there exists a countable set {f n } n∈N ⊂ C b (X) such that the fact lim D π ∆ (f n |y) = µ(f n ), for all n ∈ N, yields the convergence in question.Thus, as the set A mentioned above one can take the intersection of all A fn .Proposition 1.7.The set G is non-void.
Proof.In view of Corollary 1.5, to get the property in question it is enough to show that P y = ∅ for some y ∈ X.As the spaces X are Polish, for every ε > 0, one finds a compact A Thereby, for Λ ∈ L fin , we set Furthermore, we write Clearly, C ε is compact.Now let y ∈ C ε Λ for some Λ ∈ L fin .By (1.9), it follows that, for any and z ∈ X, (1.17) Integrating both sides of this estimate by (1.12) we get for ∆ ∈ L fin containing this which can be iterated to the following one This yields the relative weak compactness of the sequence {π ∆ (•|y)} D and hence completes the proof.
Along with the measures (1.9) we will use the following ones For V = 0, every π Λ (•|y) is independent of y and the family {π Λ } Λ∈L fin is consistent in the Kolmogorov sense.Then by the Kolmogorov lemma, G(0) is a singleton.We are going to prove that, for sparse graphs, the set G(V ) contains at most one element if V is nonzero but V is small.Here Our main result is as follows: Theorem 1.8.Let the graph be sparse.Then there exists κ * ∈ (0, 1) such that the set G(V ) contains one element only if κ(V ) κ * .
The proof is based on an extension and refinement of the Bassalygo-Dobrushin technique, developed below in a sequence of lemmas.
Lemma 2.2 (Main).Let Λ be proper in ∆ for some 0 ∈ ∂ L ∆, and let y, z ∈ X differ at this 0 only.Then The proof of this lemma is based on the following technicalities.First, by the triangle inequality, it follows that, for α, β ∈ R, Similarly, see Assertion 2 in [2], for any n ∈ N and real α 1 , . . ., α n , one has Let (Y, B(Y ), P ) be a probability space and u, v be positive measurable functions on Y .Then, c.f. (1.5), u(y)P (dy) v(y)P (dy) Now let u, v, w be positive measurable functions and w be such that P (w) = 1.Set and suppose that M − (u, v) > 0. Then w(y)u(y)P (dy) v(y)P (dy) (2.9) Let y 0 ∈ Y be such that v(y 0 ) v(y)P (dy), (2.10) which obviously exists.Suppose that there exist positive γ, ε, δ, such that Then w(y)u(y)P (dy) v(y)P (dy) where Q(dy) = v(y)P (dy)/P (v).Thereafter, the estimate (2.11) is straightforward.
Proof of Lemma 2.2: The measure (1.18) can be written in the following form Since the configuration on ∂ L ∆ \ { 0 } is going to be fixed by the very end of the proof and the concrete choice of the single-point measures plays no role, we can change χ's to χ's, which is equivalent to considering the case of ∂ L ∆ = { 0 }.Therefore, in this section we deal with the following finite graph (2.12) The proof of the lemma is based on an inductive construction employing transformations of G.
Given n, q ∈ N, let G n,q be the family of all finite graphs such that n n for all vertices, and the number of the vertices of degree n is q.We still suppose that the graphs have no loops, multiple edges, and isolated vertices.By G 2,0 we denote the family of minimal graphs, i.e., such that n = 1 for all vertices.As we are going to compare different graphs, by the end of this section we write ν G and S G instead of ν ∆ and S ∆ respectively.Finally, for a set of vertices Λ and ∈ Λ, we write Given G ∈ G 2,0 , let 1 be the only neighbor of 0 .Then the projection of the measure (1.18) onto B(X Λ ) can be written in the form where Given n, q ∈ N, suppose now that, for any G ∈ G n,q having the form (2.12), the property stated in Lemma 2. The following three cases: b ∈ Λ, b = 0 , b ∈ ∆ \ Λ, will be considered separately.
We set, c.f. (1.18) and (2.13), where Here we consider the case where b and 0 are not adjacent.The construction in the opposite case is quite similar.The projection of the measure (2.15) onto B(X Λ) can be written in the form whereas its projection onto B(X Γ ) is where Φ(x b1 , . . .x bn |y) With the help of these functions, the projection of the 'old' measure onto B(X Λ ) can be written Now by means of (2.6) one gets where (2.24) In order to proceed further in estimating A and B let us compare S G ( , 0 ) with S G ( , 0 ).As the potentials V for , ∈ ∆ aug b are the same in both measures (1.18) and (2.15), the quantities under comparing are calculated by (2.2), (2.3) with one and the same κ .Furthermore, by (2.15) it follows that Thus, taking into account that n bi = 1, i = 1, . . ., n and n b > 1, one gets, see (2.2) (2.25) Furthermore, every self-avoiding path ϑ( , 0 ) in ∆aug which starts at ∈ Λ b always avoids Γ.Therefore, Here and in the sequel, for a ∈ ∆ aug , by S a G ( , ) we denote the quantity calculated according to (2.3) with the summation taken over all paths which avoid a.The latter two estimates yield hence, Λ is proper in ∆ for 0 due to the same property of Λ in ∆ with respect to 0 .Hence, as a subset of Λ, by (2.25) Γ is also proper in ∆ for 0 .Thereafter, by the inductive assumption and (2.23), (2.24) we get Furthermore, as in (2.31) we get holding for any x b , ξ b ∈ X b .Finally, we pick ξ b ∈ X b such that Thereby, we fix x Λ ∈ X Λ , y 0 , z 0 ∈ X 0 and obtain by (2.11) that which completes the proof for this subcase.
Subcase III b:

.44)
As above, we fix x Λ ∈ X Λ , y 0 , z 0 ∈ X 0 , take arbitrarily ξ b ∈ X b , and introduce the following functions

A dense graph
Here we consider an example of the dense graph mentioned in the introduction.The graph is an expanding tree; its set of vertices is and ∆ n = ∪ n s=0 L s .Therefore, ∂ L ∆ n = L n+1 .Let ν + ∆n , n ∈ N, be the measure (1.18) corresponding to y = (y ) ∈L such that y = +1 for all ∈ L. Let also σ n be the projection of ν + ∆n onto B(X b ).To simplify notations, we write σ n (ξ) = σ n ({ξ}), ξ = ±1.Our aim is to show that, for arbitrary a > 0, lim n→+∞ σ n (1) σ n (−1) > 1, ( which means that the corresponding limiting Gibbs field1 is not invariant under the change of signs x → −x , for all ∈ L; hence, G is never singleton.To this end we construct the following recurrence.Given non-negative integer s, let us take some ∈ L s and then set Λ s = { }, and For n s, we let By σ s n , s = 0, 1, . . ., n + 1, we denote the projection of the measure ν Λ with the boundary condition y = +1 for all ∈ Λ n+1 .Then σ n+1 n (ξ) = δ ξ,1 and σ 0 n = σ n .Thereafter, the recurrence in