Convergence of rescaled "true" self-avoiding walks to the T\'oth-Werner "true" self-repelling motion

TL;DR

This paper proves that rescaled true self-avoiding walks converge to the true self-repelling motion, using a joint generalized Ray-Knight theorem to establish the weak convergence of the processes.

Contribution

It introduces a novel approach by inverting the joint generalized Ray-Knight theorem to prove the convergence of self-avoiding walks to self-repelling motion.

Findings

Rescaled true self-avoiding walks converge weakly to the true self-repelling motion.

The proof employs a joint generalized Ray-Knight theorem for local times.

The method establishes a functional limit theorem via process inversion.

Abstract

We prove that the rescaled ``true'' self-avoiding walk $(n^{-2/3}X_{\lfloor nt \rfloor})_{t\in\mathbb{R}_+}$ converges weakly as $n$ goes to infinity to the ``true'' self-repelling motion constructed by T\'oth and Werner. The proof features a joint generalized Ray-Knight theorem for the rescaled local times processes and their merge and absorption points as the main tool for showing both the tightness and convergence of the finite dimensional distributions. Thus, our result can be seen as an example of establishing a functional limit theorem for a family of processes by inverting the joint generalized Ray-Knight theorem.