Weakly self-avoiding walk is obtained by giving a penalty for every self-intersection to simple random walk path. The Edwards model is obtained by giving a penalty proportional to the square integral of the local times to the Brownian motion path, thereby also reducing the amount of time Brownian motion spends in self-intersections. We study these models in dimension one.
We prove that as the self-repellence penalty tends to zero, the large deviation rate function of the weakly self-avoiding walk converges to the rate function of the Edwards model. This shows that the speeds of one-dimensional weakly self-avoiding walk (if it exists) converges to the speed of the Edwards model. The results generalize results earlier proved only for nearest-neighbor simple random walks. The proof only uses weak convergence together with properties of the Edwards model, avoiding the rather heavy functional analysis that was used previously.
The method of proof is quite flexible, and also applies to the strictly self-avoiding case where the variance diverges. This result proves an old conjecture by Aldous.
This is joint work with Frank den Hollander and Wolfgang Koenig.