Following Frohlich and Spencer, we study one dimensional Ising spin systems with ferromagnetic, long range interactions which decay as vertical bar x-y vertical bar(-2+alpha), 0 <=alpha <= 1/2. We introduce a geometric description of the spin configurations in terms of triangles which play the role of contours and for which we establish Peierls bounds. This in particular yields a direct proof of the well-known result by Dyson about phase transitions at low temperatures. (C) 2005 American Institute of Physics

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy.In contrast with other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalisation.We also solve the associated spectral problem and explicity contruct action-angle variables through the r-matrix approach.

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In this paper we study a continuum version of the Potts model, where particles are points in R d , d ≥ 2, with a spin which may take S ≥ 3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ −1 , γ > 0. In mean field, for any inverse temperature β there is a value of the chemical potential λ β at which S + 1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, [9], we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper, [8], we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S + 1 extremal DLR measures.

Abstract. In this paper we study the statistics of combinatorial partitions of the integers, which arise when studying the occupation numbers of loops in the mean field Bose gas. We review the results of Lewis and collaborators [10] [2] and get some more precise estimates on the behavior at the critical point (fluctuations of the condensate component, finite volume corrections to the pressure). We then prove limit shape theorems for the loops occupation numbers. In particular we prove that in a certain range of the parameters, a finite fraction of the total mass is, in the limit, supported by infinitely long loops. We also show that this mass is equal to the mass of the condensed state where all particles have zero momentum. ---------------------