Application of the O(N)-hyperspherical harmonics to the study of the continuum limits of one-dimensional sigma-models and to the generation of high-temperature expansions in higher dimensions

In this talk we present the exact solution of the most general one-dimensional O(N)-invariant spin model taking values in the sphere S-N-1, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed.