ABSENCE OF HYPERSCALING VIOLATIONS FOR PHASE-TRANSITIONS WITH POSITIVE SPECIFIC-HEAT EXPONENT

Finite size scaling theory and hyperscaling are analyzed in the ensemb le limit which differs from the finite size scaling limit. Different s caling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of stat istical mechanics. This identification combined with finite ensemble s caling leads to the conclusion that hyperscaling relations cannot be v iolated for phase transitions with strictly positive specific heat exp onent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critica l point. The analytical expressions are given in terms of general H-fu nctions, scaling dimensions and a new universal shape parameter. The u niversal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behav iour. The critical finite size scaling functions for the order paramet er distribution are evaluated numerically for the cases delta=3, delta =5 and delta=15 where delta is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical predicti on and Monte-Carlo simulations for the two dimensional Ising model. An alytical expressions for critical amplitude ratios are derived in term s of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the patholog ical behaviour of the renormalized coupling constant in mean field the ory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.