Universality conjecture for random-cluster divide-and-colour scaling limits

Let X~(p,q,r)\tilde X(p,q,r) be the divide-and-colour model associated with a random-cluster measure with parameters q>0q>0 and p<pc(q)p<p_c(q), and consider it on the triangular lattice at r=1/2r=1/2. Universality conjecture. For every q>0q>0 and every p<pc(q)p<p_c(q), the site-percolation scaling limit of

X~(p,q,1/2)\tilde X(p,q,1/2)

is the same as the scaling limit of critical Bernoulli site percolation. This conjecture expresses the expectation that short-range correlations below the random-cluster critical point do not change the critical scaling limit; the source does not state a resolution.

Sources & referencesView supporting material

Primary source

Andras Balint, Federico Camia and Ronald Meester, “Sharp phase transition and critical behaviour in 2D divide and colour models”, arXiv:0708.3349 (2007).

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