Critical temperatures of continuous spin models and the free energy of a polymer

The critical temperature Tc(n,d) of a classical n-component spin model with a general continuous spin distribution on a d-dimensional hypercubic lattice is expanded in inverse powers of d to order 1/d3. The general result differs significantly from the special case of fixed spin length owing to changes in the graphical structure of the high temperature expansion. In the limit n→0 the model becomes identical to the general self-interacting random walk or polymer problem: The Boltzmann factors for self-intersections in the walks correspond to the moments of the spin distribution functions, and Tc(0,d) yields the polymer free energy.