Interface Models and Boundary Conditions

We study certain aspects of the thermodynamic formalism of interface models. Under appropriated conditions, we prove a conjecture proposed by Spohn [1] few years ago, and as a consequence the validity of exact results on the equilibrium shape associated to certain Solid-On-Solid models.     

On classical ferromagnets with a complex external field

We prove analyticity properties of correlation functions using correlation inequalities.     

Correlation functions of a lattice system

We deduce an integral equation for the infinite volume correlation functions of a class of lattice systems and we apply it to find results on the analyticity in the interaction potentials of the pressure and of the correlation functions and on the ergodicity of the equilibrium states in the gaseous phase. By similar methods we prove some cluster properties for the correlation functions in the gaseous phase.     

On the statistical mechanics of the gauge invariant Ising model

Some results on the phase structure of the gauge invariant Ising model are derived by using convergent expansions.     

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 Antonov's rule, _ab = _af + _fb , where 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.Laboratoire Propre du CNRS: LP 7061     

Correlation functions and boundary conditions in the ising ferromagnet

A number of new results on the Ising ferromagnet are obtained as a consequence of correlation inequalities. These results concern the monotonicity properties of the correlation functions, the study of equilibrium states for certain boundary conditions, and the uniqueness of the state in a semiinfinite lattice.     

On the uniqueness of the Invariant Equilibrium State and surface tension

Symmetric Equilibrium States and their properties under duality transformation are investigated. Necessary and sufficient conditions are derived for equilibrium states to be transformed into equilibrium states by duality. It is shown that ferromagnetic systems satisfying those conditions have correlation functions bounded by those corresponding to the (+) and free boundary conditions. It is then proved than any Invariant Equilibrium State of a ferromagnetic system is transformed into an equilibrium state by duality and is thus unique if the states defined by the (+), and free boundary conditions coincide on the symmetric algebra. The existence of surface tension between two pure phases is established.     

Statistical mechanics of lattice systems

We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a large set of potentials and temperatures.On partial leave from the University of Aix-Marseille.     

Statistical mechanics of quantum mechanical particles with hard cores

The states of a quantum mechanical system of hard core particles are characterized as a convex weak *compact subset of the states over aC* algebra associated with the canonical (anti-) commutation relations. It is shown that the mean conditional entropy, i.e. entropy minus energy, can be defined as an affine upper semi-continuous function over theG-invariant hard core states whereG is an invariance group containing space translations. An abstract definition of the pressure and equilibrium states is given in terms of the maximum of the conditional entropy and it is shown that the pressureP _S obtained in this way satisfiesPP _S P whereP andP are the thermodynamic pressures obtained from the usual Gibbs formalism with elastic wall, and repulsive wall, boundary conditions respectively. A number of additional results concerning the equilibrium states are also given.     

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.