Unwrapped two-point functions on high-dimensional tori

TL;DR

This paper investigates unwrapped two-point functions for various models on high-dimensional tori, revealing mean-field behavior and universal properties, contrasting with the anomalous plateau observed in standard two-point functions.

Contribution

It introduces the concept of unwrapped two-point functions for high-dimensional models and connects their asymptotic behavior to random walk models, providing new insights into their universal properties.

Findings

Unwrapped two-point functions exhibit mean-field behavior.

Asymptotics of unwrapped functions relate to random walk models.

Ising and SAW walk lengths show universal behavior on high-dimensional tori.

Abstract

We study unwrapped two-point functions for the Ising model, the self-avoiding walk and a random-length loop-erased random walk on high-dimensional lattices with periodic boundary conditions. While the standard two-point functions of these models have been observed to display an anomalous plateau behaviour, the unwrapped two-point functions are shown to display standard mean-field behaviour. Moreover, we argue that the asymptotic behaviour of these unwrapped two-point functions on the torus can be understood in terms of the standard two-point function of a random-length random walk model on Zd. A precise description is derived for the asymptotic behaviour of the latter. Finally, we consider a natural notion of the Ising walk length, and show numerically that the Ising and SAW walk lengths on high-dimensional tori show the same universal behaviour known for the SAW walk length on the complete graph.