Abstract

We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion (Ann. of Math. (2) 189:1 (2019), 75–99) to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin–Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov (Dokl. Akad. Nauk 288:6 (1986), 1308–1311) but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following:

1. The critical exponent inequalities $\gamma \le \delta -1$ and $\Delta \le \gamma +1$ hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on ${\mathbb{Z}}^d$, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with $q\in \left[1,2\right)$.
2. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.

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