Geodesic
Planar random-cluster model: scaling relations
Forum of Mathematics, Pi, Tome 10 (2022)

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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 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 $\iota $ replacing the four-arm event exponent $\xi _4$. 
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Hugo Duminil-Copin; Ioan Manolescu. Planar random-cluster model: scaling relations. Forum of Mathematics, Pi, Tome 10 (2022). doi: 10.1017/fmp.2022.16

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