Salvador Miracle-Sole

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.

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd⩾2) undergoes a first-order phase transition as the inverse temperatureβ is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpΔE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted byΔβm(L)=(Inq/ΔE)L−d+O(L−2d) with respect to the infinite-volume transition pointβt. We also propose an alternative definition of the finite-volume transition temperatureβt(L) which might be useful for numerical calculations because it differs only by exponentially small corrections fromβt.

Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model

We derive new inequalities for the plane rotator ferromagnetic model and use them to obtain the following results:1)If the model is isotropic, the derivability of the free energy as function of the magnetic fieldh implies the existence of a unique translation invariant Gibbs state and if furthermoreh=0 all Gibbs states are invariant by rotation of the spins.2)If the model is anisotropic the above assertion holds forh non-zero.3)If the model is anisotropic then there are at most two extremal translation invariant Gibbs states for almost all values of the anisotropy parameter.

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 convergence of cluster expansions

Two theorems on the theory of cluster expansions for an abstract polymer system are reported.

Surface tension, step free energy, and facets in the equilibrium crystal

Some aspects of the microscopic theory of interfaces in classical lattice systems are developed. The problem of the appearance of facets in the (Wulff) equilibrium crystal shape is discussed, together with its relation to the discontinuities of the derivatives of the surface tension τ(n) (with respect to the components of the surface normaln) and the role of the step free energy τstep(m) (associated with a step orthogonal tom on a rigid interface). Among the results are, in the case of the Ising model at low enough temperatures, the existence of τstep(m) in the thermodynamic limit, the expression of this quantity by means of a convergent cluster expansion, and the fact that 2τstep(m) is equal to the value of the jump of the derivative ∂τ/∂δ (when δ varies) at the point δ=0 [withn=(m1 sin δ,m2 sin δ, cos δ)]. Finally, using this fact, it is shown that the facet shape is determined by the function τstep(m).

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.

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.

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