
SCALING ABOVE THE UPPER CRITICAL DIMENSIONBertrand Berche (Personal webpage)IJL, University de Lorraine
Ralph Kenna (Personal webpage)Applied Mathematics Research Centre, Coventry University
Above the upper critical dimension, the breakdown of hyperscaling is associated with dangerous irrelevant variables in the renormalization group formalism at least for systems with periodic boundary conditions. While these have been extensively studied, there have been only a few analyses of finitesize scaling with free boundary conditions. The conventional paradigm there is that, in contrast to periodic geometries, finitesize scaling is Gaussian, governed by a correlation length comensurate with the lattice extent. Here, we present analytical and numerical results which indicate that this paradigm is unsupported, both at the infinitevolume critical point and at the pseudocritical point where the finitesize susceptibility peaks. Instead the evidence indicates that finitesize scaling at the pseudocritical point is similar to that in the periodic case. An analytic explanation is offered which allows hyperscaling to be extended beyond the upper critical dimension. 