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It is known that the random cluster model with $q = 1$ corresponds to bond percolation, and $q = 2, 3, ... $ corresponds to the $q$-state Potts model. Both of these have a local description.

What about a random cluster model with non-integer $q$?

I think the answer is no, and I suppose there is a way to see this by computing certain correlations. Does anyone know a proof?

https://en.wikipedia.org/wiki/Random_cluster_model

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  • $\begingroup$ This question should be made more precise (but I don't know how). For instance, I have recently been reminded that the random cluster model with $q\in [4,\infty)$ admits a local description (through a mapping to the 8-vertex model) at the self-dual point (which corresponds to the phase transition point). This particular mapping does not seem to lead to a local description away from this point, but this still shows that things may be somewhat tricky. $\endgroup$
    – Yvan Velenik
    Commented Oct 19, 2023 at 9:23

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