The strong interaction limit of continuous-time weakly self-avoiding walk

TL;DR

This paper investigates the strong interaction limit of continuous-time weakly self-avoiding walks, revealing a dependence on fugacity scaling and introducing a new 'quick step' model that interpolates between known behaviors.

Contribution

It clarifies the limiting behavior of continuous-time weakly self-avoiding walks and introduces the 'quick step' model as an interpolation between self-avoiding walk and simple random walk.

Findings

Limit depends on fugacity scaling.

Introduces 'quick step' model as an interpolation.

Analyzes limits for fixed steps and two-point function.

Abstract

The strong interaction limit of the discrete-time weakly self-avoiding walk (or Domb--Joyce model) is trivially seen to be the usual strictly self-avoiding walk. For the continuous-time weakly self-avoiding walk, the situation is more delicate, and is clarified in this paper. The strong interaction limit in the continuous-time setting depends on how the fugacity is scaled, and in one extreme leads to the strictly self-avoiding walk, in another to simple random walk. These two extremes are interpolated by a new model of a self-repelling walk that we call the "quick step" model. We study the limit both for walks taking a fixed number of steps, and for the two-point function.