Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model

The two-dimensional Potts model can be studied either in terms of the original Q-component spins, or in the geometrical reformulation via Fortuin-Kasteleyn (FK) clusters. While the FK representation makes sense for arbitrary real values of Q by construction, it was only shown very recently that the spin representation can be promoted to the same level of generality. In this paper we show how to define the Potts model in terms of observables that simultaneously keep track of the spin and FK degrees of freedom. This is first done algebraically in terms of a transfer matrix that couples three different representations of a partition algebra. Using this, one can study correlation functions involving any given number of propagating spin clusters with prescribed colours, each of which contains any given number of distinct FK clusters. For 0 <= Q <= 4 the corresponding critical exponents are all of the Kac form h_{r,s}, with integer indices r,s that we determine exactly both in the bulk and in the boundary versions of the problem. In particular, we find that the set of points where an FK cluster touches the hull of its surrounding spin cluster has fractal dimension d_{2,1} = 2 - 2 h_{2,1}. If one constrains this set to points where the neighbouring spin cluster extends to infinity, we show that the dimension becomes d_{1,3} = 2 - 2 h_{1,3}. Our results are supported by extensive transfer matrix and Monte Carlo computations.Comment: 15 pages, 3 figures, 2 table

Critical domain walls in the Ashkin-Teller model

We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE$_\kappa$, except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE$_\kappa$ in this case.Comment: Final version published in JSTAT. 13 pages, 5 figures. Substantial changes in the data production, analysis and in the conclusions. Added a section about the crossing probability. Typeset with 'iopart