On the Kertész line: Some rigorous bounds

Abstract

We study the Kertész line of the q-state Potts model at (inverse) temperature β in the presence of an external magnetic field h. This line separates the 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.