Jean Ruiz

On the Purity of the Limiting Gibbs State for the Ising Model on the Bethe Lattice

We give a proof that for the Ising model on the Bethe lattice, the limiting Gibbs state with zero effective field (disordered state) persists to be pure for temperature below the ferromagnetic critical temperatureTcF until the critical temperatureTcSG of the corresponding spin-glass model. This new proof revises the one proposed earlier.

Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation

We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states. We use the Pirogov-Sinaï theory which we adapt to the Fortuin-Kasteleyn representation of the models. This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities. As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions. This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point.

Phases coexistence and surface tensions for the potts model

Theq states Potts model exhibits a first order phase transition at some inverse temperature βt between “ordered” and “disordered” phases forq large as proved in [1]. In space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at βt to derive an estimate on the probability of a contour. This enables us to prove the preceding result and the following new results:(i)The discontinuity of the mass gap at βt.(ii)The existence of astrictly positive surface tension between two ordered phases up to βt.(iii)The existence of a non-zero surface tension between an “ordered” and the “disordered” phase at βt.

Convexity properties of the surface tension and equilibrium crystals

We study the thermodynamic limit of the orientation-dependent surface tension. Under general conditions, which we show to hold true for a large class of lattice systems, we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by R. L. Dobrushin and S. B. Shlosman(1). We discuss some consequences of these results for the equilibrium crystal shape.

Theq-state Potts model in the standard Pirogov-Sinai theory: Surface tensions and Wilson loops

Theq-state Potts model (both scalar and gauge versions) is rewritten, with the help of the duality transformation, into a form of the Pirogov-Sinai theory with noninteracting contours that can be controlled by cluster expansions onceq is large enough. This is then used in a new proof of the existence of a unique transition (inverse) temperatureβt, where the mean internal energy is discontinuous. Moreover, we prove for the scalar model (again forq large enough) that there are discontinuities atβt of the magnetization and of the mass gap, with the magnetization vanishing belowβt and the mass gap vanishing aboveβt. We also show that the surface tensions between ordered stable phases are strictly positive up toβt, and the surface tension between an ordered phase and the disordered one is strictly positive atβt. For the three-dimensional gauge model, the Wilson parameter exhibits a direct transition from an area law decay (quark confinement) to a perimeter law decay (deconfinement).

Interfaces in the Potts Model II: Antonov's Rule and Rigidity of the Order Disorder Interface

Within the ferromagneticq-state Potts model we discuss the wetting of the interface between two ordered phasesa andb by the disordered phasef at the transition temperature. In two or more dimensions and forq large we establish the validity of the Antonovs rule, σab = σaf + σfb, where σ denotes the surface tension between the considered phases. We also prove that at this temperature, in three or more dimensions the interface between any ordered phase and the disordered one is rigid.

On a Model of Random Cycles

Abstract We consider a model of random permutations of the sites of the cubic lattice. Permutations are weighted so that sites are preferably sent onto neighbors. We present numerical evidence for the occurrence of a transition to a phase with infinite, macroscopic cycles.

A Study of Perfect Wetting for Potts and Blume-Capel Models with Correlation Inequalities

AbstractThe so-called perfect wetting phenomenon is studied for theq-state,d⩾2 Potts model. Using a new correlation inequality, a general inequality is established for the surface tension between ordered phases (σa,b) and the surface tension between an ordered and the disordered phases (σa,f) for any even value ofq. This result implies in particular

ON FREE ENERGIES OF THE ISING MODEL ON THE CAYLEY TREE

\sigma _{\beta _l }^{a,b} \geqslant \sigma _{\beta _l }^{a,f} + \sigma _{\beta _l }^{b,f} > 0