On the Kertész line: Some rigorous bounds

We study the Kertész line of the q–state Potts model at (inverse) temperature β, in presence of an external magnetic field h. This line separates two regions of the phase diagram according to the existence or not of an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is known that the Kertész line hK(β) coincides with the line of first order phase transition for small fields when q is large enough. Here we prove that the first order phase transition implies a jump in the density of the infinite cluster, hence the Kertész line remains below the line of first order phase transition. We also analyze the region of large fields and prove, using techniques of stochastic comparisons, that hK(β) equals log(q−1)−log(β−βp) to the leading order, as β goes to βp = − log(1−pc) where pc is the threshold for bond percolation. One important feature of the Fortuin–Kasteleyn representation of Ising and Potts models [1] (the random cluster model), is that the geometrical transition, i.e. the apparition of an infinite cluster, corresponds precisely to the phase transition leading to a spontaneous magnetization in the absence of an external field [2]. In [3], Kertész pointed out that this property is lost in the Ising model when an external field h is introduced: while thermodynamic quantities are analytic for any h > 0, a geometric transition appears in the corresponding random cluster model and there is a whole percolation transition line extending from the Curie point (h = 0) to infinite fields. As Kertész explained, the analyticity of thermodynamic quantities and the existence of the percolation transition are not contradictory because the free energy remains analytic. This document has been produced using TEXmacs(see http://www.texmacs.org)