Geometric explanation of anomalous finite-size scaling in high dimensions

TL;DR

This paper provides a geometric perspective explaining why standard finite-size scaling appears to fail in high-dimensional systems, and proposes a correction by redefining correlation functions to account for windings.

Contribution

It introduces a geometric explanation for anomalous scaling in high dimensions and demonstrates how to correct it by adjusting the correlation function scale.

Findings

Anomalous behavior in correlation functions can be corrected by accounting for windings.

Simulations on five-dimensional lattices support the geometric explanation.

Redefining correlation functions on a winding-aware scale restores expected scaling behavior.

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.