We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion (Ann. ofMath. (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:
The critical exponent inequalities
and
hold for percolation and the random cluster model on any transitive graph.
These inequalities are new even in the context of Bernoulli percolation on
,
and are saturated in mean-field for Bernoulli percolation and for the
random cluster model with
.
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.