Phase transition in continuum Potts models

Abstract

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with superstable interaction.

A Phase Transition in a Quenched Amorphous Ferromagnet

“…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.…”

“…The probability distribution of the random graph (γ , ε γ ) is constructed in the following way, cf. [17]. Let ε denote a set of pairs of distinct points, i.e., of e = {x, y}, x, y ∈ R d , x = y.…”

Mean-field theory of the Potts gas

“…The first model of this type was introduced by Widom and Rowlinson [12]; having q = 2 colours and hard-core intercolour repulsion. A colour-ordering transition for large activity has been established for this model by Ruelle [13], and for its soft-core counterpart by Lebowitz and Lieb [14]; see also the later studies by Bricmont, Kuroda and Lebowitz [5], Chayes, Chayes and Kotecky [15], and Georgii and Häggström [16]. No rigorous results, however, are available on the system's behaviour at the activity threshold to colour-ordering -in fact, even the existence of such a threshold is unknown -, and only numerical results are available [17,18,19].…”

Gibbs states of continuum particle systems with unbounded spins: Existence and uniqueness

“…The corresponding physical systems are e.g. magnetic gases, ferrofluids, amorphous magnets, etc., see [16], [17], [36]. Such compound (with additional spin variables) models are of a special interest in mathematical physics because they provide some (of still very few) examples of continuum systems where the appearence of an (orientational odering) phase transition has been proved rigorously.…”

“…The question of the existence of multipliple Gibbs states (phase transitions) has been discussed for ferromagnetic interactions in [39], [16], [5] (discrete spins), [17], [36] (hard core position-position interaction, continuous scalar spins) and in our complementary paper [9] (no hard core, continuous scalar spins). The appearance of Berezinskii-Kosterlitz-Thouless phase transition in a ferrofluid of hard-core particles with O(2)-invariant spins was shown in [18], see also references given there.…”