Finite-size Scaling and Surface Tension from Effective One-dimensional Systems

We develop a method for precise asymptotic analysis of partition functions near firstorder phase transitions. Working in a (ν+1)-dimensional cylinder of volume L×. . .×L×t, we show that leading exponentials in t can be determined from a simple matrix calculation provided t ≥ ν logL. Through a careful surface analysis we relate the off-diagonal matrix elements of this matrix to the surface tension and L, while the diagonal matrix elements of this matrix are related to the metastable free energies of the model. For the off-diagonal matrix elements, which are related to the crossover length from hypercubic (L = t) to cylindrical (t = ∞) scaling, this includes a determination of the pre-exponential power of L as a function of dimension. The results are applied to supersymmetric field theory and, in a forthcoming paper, to the finite-size scaling of the magnetization and inner energy at field and temperature driven first order transitions in the crossover region from hypercubic to cylindrical scaling. † Including the corrections made while proof reading for Commun. Math. Phys. ∗ Research partially supported by the A. P. Sloan Foundation and by the NSF under DMS-8858073. ∗∗ Research partially supported by the NSF under DMS-8858073 and DMS-9008827. 1