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

TL;DR

This paper proves that the escape speed of a continuous-time weakly self-avoiding walk on the integers increases strictly with the repelling strength, using advanced mathematical techniques.

Contribution

It establishes the strict monotonicity of the escape speed in the continuous-time WSAW model, providing a new proof and extending previous results.

Findings

Escape speed is strictly increasing with repelling strength.

New proof of existence of the escape speed.

Application of supersymmetric transfer matrix method.

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.