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 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.

arXiv comment5 pages, 4 figures

References in corpus (15)

• Worm algorithms for classical statistical models
• Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
• On the Number of Incipient Spanning Clusters
• A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
• Hyperscaling above the upper critical dimension
• Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
• Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model
• Finite-size scaling above the upper critical dimension
• On the role of Fourier modes in finite-size scaling above the upper critical dimension
• Non-vanishing boundary effects and quasi-first order phase transitions in high dimensional Ising models
• Fisher's scaling relation above the upper critical dimension
• Finite size scaling of the 5D Ising model with free boundary conditions
• Scaling of Self-Avoiding Walks in High Dimensions
• The scaling window of the 5D Ising model with free boundary conditions
• Irreversible Markov chain Monte Carlo algorithm for self-avoiding walk

Cited by in corpus (21)

• Critical Casimir Effect: Exact Results
• Extraordinary-log surface phase transition in the three-dimensional $XY$ model
• Random-length Random Walks and Finite-size scaling in high dimensions
• Universal Finite-Size Scaling for Percolation Theory in High Dimensions
• Finite-size Scaling of O($n$) Systems at the Upper Critical Dimensionality
• Geometric Upper Critical Dimensions of the Ising Model
• Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Lifted Worm Algorithm for the Ising Model
• The length of self-avoiding walks on the complete graph
• Quantum extraordinary-log universality of boundary critical behavior
• Self-similarity breaking: Anomalous nonequilibrium finite-size scaling and finite-time scaling
• Classical-quantum correspondence of special and extraordinary-log criticality: Villain's bridge
• Logarithmic finite-size scaling of the self-avoiding walk at four dimensions
• Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph
• Non-equilibrium relaxations: ageing and finite-size effects
• Unwrapped two-point functions on high-dimensional tori
• Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions
• Ising model with Curie-Weiss perturbation
• Near-critical and finite-size scaling for high-dimensional lattice trees and animals
• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model