I will discuss the nature of the low-temperature expansion for the multipoint spin correlations of classical O(N) vector models in dimension three and higher. The main result I will present is that such expansions define asymptotic series, with explicit bounds 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. The method of proof generalizes an approach, proposed originally by Bricmont-Fontaine-Lebowitz-Lieb-Spencer in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables. Time permitting, I will comment on the perspectives and open problems for quantum spin systems. Joint work with Sebastien Ott.
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