Abstract. We study the surface tension and the phenomenon of phase coexistence for the Ising model on Z d (d 2) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations : upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value τ q of surface tension to maximal flows (first passage times if d = 2). For a broad class of distributions of the couplings we show that the inequality τ a τ q -where τ a is the surface tension under the averaged Gibbs measure -is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models.
By means of a multi-scale analysis we describe the typical geometrical structure of the clusters under the FK measure in random media. Our result holds in any dimension d 2 provided that slab percolation occurs under the averaged measure, which should be the case for the whole supercritical phase. This work extends that of Pisztora [A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (4) (1996) 427-466] and provides an essential tool for the analysis of the supercritical regime in disordered FK models and in the corresponding disordered Ising and Potts models.
We present a study of phase transitions of the mean-field Potts model at (inverse) temperature β, in presence of an external field h. Both thermodynamic and topological aspects of these transitions are considered. For the first aspect we complement previous results and give an explicit equation of the thermodynamic transition line in the β-h plane as well as the magnitude of the jump of the magnetization (for q 3). The signature of the latter aspect is characterized here by the presence or not of a giant component in the clusters of a Fortuin-Kasteleyn type representation of the model. We give the equation of the Kertész line separating (in the β-h plane) the two behaviours. As a result, we get that this line exhibits, as soon as q 3, a very interesting cusp where it separates from the thermodynamic transition line.
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