Random Curves and the Antiferromagnetic Transition in the Two-Dimensional Potts Model
2007, v.13, Issue 4, 637-658
We review our recent results on the antiferromagnetic transition in the two-dimensional Potts model and its connection to random curves. This transition presents several unusual features. First, it is a so-called first-order critical point, i.e., it is simultaneously first and second order. Second, its Coulomb gas consists of two fields, one of which is non-compact, i.e., it contributes a continuous part to the spectrum of critical exponents. Third, whenever $Q$ is a Beraha number, i.e., $Q=4 \cos^2 (\pi/p)$ with $p$ integer, the physics is profoundly different from the case of generic $Q$. The limit $Q\to 0$ is equivalent to a combinatorial model of unrooted spanning forests, which we describe in some detail.
Keywords: Potts model,antiferromagnetism,random curves,spanning forests