  We prove rigorously that the ferromagnetic Ising model on any nonamenable
Cayley graph undergoes a continuous (second-order) phase transition in the
sense that there is a unique Gibbs measure at the critical temperature. The
proof of this theorem is quantitative and also yields power-law bounds on the
magnetization at and near criticality. Indeed, we prove more generally that the
magnetization $\langle \sigma_o \rangle_{\beta,h}^+$ is a locally
H\"older-continuous function of the inverse temperature $\beta$ and external
field $h$ throughout the non-negative quadrant $(\beta,h)\in [0,\infty)^2$. As
a second application of the methods we develop, we also prove that the free
energy of Bernoulli percolation is twice differentiable at $p_c$ on any
transitive nonamenable graph.
