title: Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions.
abstract: 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 behavior for correlation functions, measured on the standard Euclidean scale, can be removed by defining correlation functions on a scale which correctly accounts for windings.
authors.affiliation: ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
authors.forename: Jens
authors.initials: J
authors.lastname: Grimm
authors.affiliation: School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
authors.forename: Eren Metin
authors.initials: EM
authors.lastname: Elçi
authors.affiliation: ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
authors.forename: Zongzheng
authors.initials: Z
authors.lastname: Zhou
authors.affiliation: ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia.
authors.forename: Timothy M
authors.initials: TM
authors.lastname: Garoni
authors.affiliation: Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230027, China and CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
authors.forename: Youjin
authors.initials: Y
authors.lastname: Deng