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The two-dimensional Potts model can be studied either in terms of theoriginal Q-component spins, or in the geometrical reformulation viaFortuin-Kasteleyn (FK) clusters. While the FK representation makes sense forarbitrary real values of Q by construction, it was only shown very recentlythat 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 observablesthat simultaneously keep track of the spin and FK degrees of freedom. This isfirst done algebraically in terms of a transfer matrix that couples threedifferent representations of a partition algebra. Using this, one can studycorrelation functions involving any given number of propagating spin clusterswith prescribed colours, each of which contains any given number of distinct FKclusters. For 0 <= Q <= 4 the corresponding critical exponents are all of theKac form h_{r,s}, with integer indices r,s that we determine exactly both inthe bulk and in the boundary versions of the problem. In particular, we findthat the set of points where an FK cluster touches the hull of its surroundingspin cluster has fractal dimension d_{2,1} = 2 - 2 h_{2,1}. If one constrainsthis set to points where the neighbouring spin cluster extends to infinity, weshow that the dimension becomes d_{1,3} = 2 - 2 h_{1,3}. Our results aresupported by extensive transfer matrix and Monte Carlo computations.

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