The scaling limit of the weakly self-avoiding walk on a high-dimensional torus

TL;DR

This paper proves that the weakly self-avoiding walk on a high-dimensional torus converges to Brownian motion under certain scaling conditions, with the diffusion constant matching that on the infinite lattice, revealing the walk's insensitivity to the torus topology up to a certain length scale.

Contribution

It establishes the scaling limit of the weakly self-avoiding walk on high-dimensional tori as Brownian motion and confirms the diffusion constant matches the infinite lattice case, highlighting the walk's local behavior.

Findings

Scaling limit is Brownian motion on the torus for walk length o(V^{1/2})

Diffusion constant matches that of the walk on ^d

Walk does not feel the torus topology up to about V^{1/2} steps

Abstract

We prove that the scaling limit of the weakly self-avoiding walk on a $d$-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1/2})$ where $V$ is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on $\mathbb{Z}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the torus up until it reaches about $V^{1/2}$ steps which we believe is sharp.