Random-length Random Walks and Finite-size scaling in high dimensions

TL;DR

This paper introduces a random-length random walk model to rigorously analyze finite-size scaling in high-dimensional Ising models, confirming universality through theoretical proofs and extensive simulations.

Contribution

It provides a rigorous proof that the random walk model exhibits the same universal finite-size scaling behavior as the Ising model and self-avoiding walk in high dimensions.

Findings

The model's mean walk length controls Green's function scaling.

Universality of finite-size scaling confirmed via Monte Carlo simulations.

Results apply to high-dimensional lattices with different boundary conditions.

Abstract

We address a long-standing debate regarding the finite-size scaling of the Ising model in high dimensions, by introducing a random-length random walk model, which we then study rigorously. We prove that this model exhibits the same universal FSS behaviour previously conjectured for the self-avoiding walk and Ising model on finite boxes in high-dimensional lattices. Our results show that the mean walk length of the random walk model controls the scaling behaviour of the corresponding Green's function. We numerically demonstrate the universality of our rigorous findings by extensive Monte Carlo simulations of the Ising model and self-avoiding walk on five-dimensional hypercubic lattices with free and periodic boundaries.