Geometric explanation of anomalous finite-size scaling in high   dimensions
  We give an intuitive geometric explanation for the apparent breakdown of standard finite-size scaling in systems with periodic boundaries above the upper critical dimension. The Ising model and self-avoiding walk are simulated on five-dimensional hypercubic lattices with free and periodic boundary conditions, by using geometric representations and recently introduced Markov-chain Monte Carlo algorithms. We show that previously observed anomalous behaviour for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings. 
Jens Grimm, Eren Metin El\c{c}i, Zongzheng Zhou, Timothy M. Garoni,   Youjin Deng
cond-mat.stat-mech