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arXiv · 2210.15580

Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed

Yucheng Liu
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Abstract

Weakly self-avoiding walk (WSAW) is a model of simple random walk paths that penalizes self-intersections. On $\mathbb{Z}$, 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.

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Yucheng Liu. 2024-07-01. Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed. https://doi.org/10.1214/24-aap2098

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