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

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