%0 Book Section %1 statphys23_0976 %A Hucht, A. %A Luebeck, S. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K correlation critical dimension finite-size length montecarlo scaling simulations statphys23 topic-2 upper %T On the breakdown of finite-size scaling in high dimensional systems %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=976 %X 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)

@incollection{statphys23_0976, abstract = {Finite-size scaling functions of continuous phase transitions exhibit a scaling anomaly above the upper critical dimension $d_\mathrm{c}$. 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 $2 \leq d \leq 10$ dimensions with various boundary conditions. For $d<d_\mathrm{c}=6$ the spatial correlation length at criticality, $\xi_\mathrm{c}$, is limited by the systems size $L$ for all BCs, whereas it exceeds the systems size as $\xi_\mathrm{c} \sim L^{d/d_c}$ in systems with periodic BC above $d_\mathrm{c}$, 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)}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Hucht, A. and Luebeck, S.}, biburl = {https://www.bibsonomy.org/bibtex/267bd0da9d00a6c391e539b5228fd116e/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {4f422aedc7587b16c99998f14c8c70be}, intrahash = {67bd0da9d00a6c391e539b5228fd116e}, keywords = {correlation critical dimension finite-size length montecarlo scaling simulations statphys23 topic-2 upper}, month = {9-13 July}, timestamp = {2007-06-20T10:16:36.000+0200}, title = {On the breakdown of finite-size scaling in high dimensional systems}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=976}, year = 2007 }