## SCALING ABOVE THE UPPER CRITICAL DIMENSION

### Bertrand Berche

##### IJL, University de Lorraine

### Ralph Kenna

##### 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 finite-size scaling with free
boundary conditions. The conventional paradigm there is that, in
contrast to periodic geometries, finite-size 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
infinite-volume critical point and at the pseudocritical point where
the finite-size susceptibility peaks. Instead the evidence indicates
that finite-size 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.

* Personal webpage (Berche) *
* Personal webpage (Kenna)*