Journal of Statistical Physics, Vol. 87. Nos. 3/4. 1997, p. 755-798.
Linked cluster expansions are generalized from an infinite to a finite volume. They are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase. A new criterion is proposed to distinguish first- from second-order transitions within a finite-size scaling analysis. The criterion applies also to other methods for investigating the phase structure, such as Monte Carlo simulations. Our computational tools are illustrated with the example of scalar O(N) models with four- and six-point couplings for N = 1 and N=4 in three dimensions. It is shown how to localize the tricritical line in these models. We indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity.