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 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 - 2h(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 - 2h(1,3). Our results are supported by extensive transfer matrix and Monte Carlo computations.