Planar random-cluster model: fractal properties of the critical phase

This paper is studying the critical regime of the planar random-cluster model on $${\mathbb {Z}}^2$$ Z 2 with cluster-weight $$q\in [1,4)$$ q ∈ [ 1 , 4 ) . More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation ($$q = 1$$ q = 1 ) and the FK-Ising model ($$q = 2$$ q = 2 ). Finally, we prove new bounds on the one, two and four-arm exponents for $$q\in [1,2]$$ q ∈ [ 1 , 2 ] , as well as the one-arm exponent in the half-plane. These improve the previously known bounds, even for Bernoulli percolation.