Abstract
Weakly self-avoiding walk (WSAW) is a model of simple random walk paths that penalizes self-intersections. On , Greven and den Hollander proved in 1993 that the discrete-time weakly self-avoiding walk has an asymptotically deterministic escape speed, and they conjectured that this speed should be strictly increasing in the repelling strength parameter. We study a continuous-time version of the model, give a different existence proof for the speed, and prove the speed to be strictly increasing. The proof uses a transfer matrix method implemented via a supersymmetric version of the BFS–Dynkin isomorphism theorem, spectral theory, Tauberian theory, and stochastic dominance.
Funding Statement
This work was supported in part by NSERC of Canada.
Acknowledgments
We thank Gordon Slade for discussions and very helpful advice. We thank Roland Bauerschmidt for introducing the problem and the transfer matrix approach to us. We thank Frank den Hollander for encouragement and comments.
Citation
Yucheng Liu. "Continuous-time weakly self-avoiding walk on has strictly monotone escape speed." Ann. Appl. Probab. 34 (6) 5522 - 5555, December 2024. https://doi.org/10.1214/24-AAP2098
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