Planar random-cluster model: scaling relations

This paper studies the critical and near-critical regimes of the planar random-cluster model on Z2 with cluster-weight ?? is an element of [1,4] using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents ??, ??, ??, ??, ??, ?? as well as ?? (when?? >= 0 ). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of the influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalisation of Kesten's classical scaling relation for Bernoulli percolation involving the "mixing rate' critical exponent??replacing the four-arm event exponent ??4.