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

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.