Planar random-cluster model: scaling relations

Abstract

This paper studies the critical and near-critical regimes of the planar random-cluster model on ${\mathbb{Z}}^2$ with cluster-weight $q\in [1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta$, $\gamma$, $\delta$, $\eta$, $\nu$, $\zeta$ as well as $\alpha$ (when $\alpha \ge 0$). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalization of Kesten's classical scaling relation for Bernoulli percolation involving the ``mixing rate'' critical exponent $\iota$ replacing the four-arm event exponent ${\xi }_4$.

85 pages, 14 figures, 1 table