Zusammenfassung
Finite-size scaling functions of continuous phase transitions exhibit a
scaling anomaly above the upper critical dimension
$d_c$. This so-called breakdown of finite-size scaling is
well-established on the basis of field theoretical and numerical
approaches for system with periodic boundary conditions (BC), both in
equilibrium (e.g. the Ising model) and non-equilibrium (e.g. directed
percolation 1). Less work was done for geometric phase transitions
and for Dirichlet BC. Therefore, we numerically investigate the bond
percolation transition in $2 d 10$ dimensions with various
boundary conditions. For $d<d_c=6$ the spatial correlation
length at criticality, $\xi_c$, is limited by the systems size
$L$ for all BCs, whereas it exceeds the systems size as
$\xi_c L^d/d_c$ in systems with periodic BC above
$d_c$, the hallmark of the breakdown of finite-size
scaling. We present, to our knowledge for the first time, a
phenomenological and descriptive interpretation of this breakdown of
finite-size scaling. Using a generalized distance definition, the correlation length $\xi$ can be directly measured in simulations, even when it exceeds the linear system size $L$.
Furthermore, we show that the high-dimensional
behavior depends strongly on the boundary conditions.
1) S. Luebeck and H.-K. Janssen, Phys. Rev. E 72, 016119 (2005)
Nutzer