  We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion 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 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 $\gamma \leq \delta-1$ and $\Delta \leq
\gamma +1$ 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 $q \in [1,2)$.
  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.
