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 where V is the volume (number of points) of the torus and if . 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 . 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 steps, which we believe is sharp.
Funding Statement
This work was supported in part by NSERC of Canada.
Acknowledgments
We thank Gordon Slade for introducing this topic to us, for our collaboration on [18] where the idea of this paper idea arose as a natural continuation as well as for useful comments on an earlier version of this paper.
Citation
Emmanuel Michta. "The scaling limit of the weakly self-avoiding walk on a high-dimensional torus." Electron. Commun. Probab. 28 1 - 13, 2023. https://doi.org/10.1214/23-ECP531
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