Abstract
This paper is studying the critical regime of the planar random-cluster model on {mathbb {Z}}^2 with cluster-weight qin [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) and the FK-Ising model (q = 2). Finally, we prove new bounds on the one, two and four-arm exponents for qin [1,2], as well as the one-arm exponent in the half-plane. These improve the previously known bounds, even for Bernoulli percolation.
Highlights
We will use standard properties of the random-cluster model
In seminal papers [6,7], Belavin, Polyakov and Zamolodchikov went even further by suggesting a much stronger invariance of statistical physics models at criticality: since the scaling limit quantum field theory is a local field, it should be invariant by any map which is locally a composition of translation, rotation and homothety, which led them to postulate full conformal invariance
One of the drawbacks of previous results is that the estimates were restricted to rectangles. This restriction is substantial in terms of applications due to the fact that boundary conditions do influence the configuration heavily in D, and that the roughness of the boundary could dictate the strength of this influence
Summary
Understanding the behaviour of physical systems undergoing a continuous phase transition at their critical point is one of the major challenges of modern statistical physics, both on the physics and the mathematical sides. In seminal papers [6,7], Belavin, Polyakov and Zamolodchikov went even further by suggesting a much stronger invariance of statistical physics models at criticality: since the scaling limit quantum field theory is a local field, it should be invariant by any map which is locally a composition of translation, rotation and homothety, which led them to postulate full conformal invariance. These papers gave birth to Conformal Field Theory, one of the most studied domains of modern physics. While we are currently unable to show existence and conformal invariance of the scaling limit for general cluster weight
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