We consider classical
vector models in dimension three and higher and investigate the nature of the
low-temperature expansions for their multipoint spin correlations. We prove that
such expansions define asymptotic series, and derive explicit estimates on
the error terms associated with their finite-order truncations. The result
applies, in particular, to the spontaneous magnetization of the 3D Heisenberg
model. The proof combines a priori bounds on the moments of the local spin
observables, following from reflection positivity and the infrared bound, with an
integration-by-parts method applied systematically to a suitable integral
representation of the correlation functions. Our method generalizes an approach,
proposed originally by Bricmont and collaborators (1981) in the context of
the rotator model, to the case of nonabelian symmetry and nongradient
observables.