FINITE-SIZE SCALING FOR POTTS MODELS

Recently, Borgs and Kotecky developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagnetic q-state Potts model, which (for q large and d greater-than-or-equal-to 2) undergoes a first-order phase transition as the inverse temperature beta is varied. We prove a formula for the internal energy in a periodic cube of side length L which describes the rounding of the infinite-volume jump DELTA-E in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by DELTA-beta-m(L) = (ln q/DELTA-E) L(-d + O(L(-2d)) with respect to the infinite-volume transition point beta-t. We also propose an alternative definition of the finite-volume transition temperature beta-t(L) which might be useful for numerical calculations because it differs only by exponentially small corrections from beta-t.