  We prove that for Bernoulli percolation on $\mathbb{Z}^d$, $d\geq 2$, the
percolation density is an analytic function of the parameter in the
supercritical interval. For this we introduce some techniques that have further
implications. In particular, we prove that the susceptibility is analytic in
the subcritical interval for all transitive short- or long-range models, and
that $p_c^{bond} <1/2$ for certain families of triangulations for which
Benjamini \& Schramm conjectured that $p_c^{site} \leq 1/2$.
