Bassalygo-Dobrushin uniqueness for continuous spin systems on irregular graphs

Abstract

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 n ∈ N, such that n` 6 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