Functional limit theorems for self-interacting random walks

Abstract

We consider a wide class of nearest neighbor self-interacting random walks on $\mathbb{Z}$ and show that, under mild conditions, their scaling limits at the process level can be uniquely identified and constructed directly from the augmented joint Ray-Knight theorem for their local times, i.e., the joint convergence of rescaled local time ``curves'' of the walks stopped at certain stopping times together with the merge and absorption times of these ``curves'' to a concatenation of coalescing reflected-absorbed diffusions and their merge and absorption times. We apply our results to polynomially self-repelling walks introduced and studied in the mid-90s by B. Tóth, and to ``true'' self-avoiding walks with bond repulsion. The first application solves a problem that has been open since Tóth's work: it gives a construction and first properties of a new family of processes, polynomially self-repelling motions, as scaling limits of polynomially self-repelling walks. These processes were simultaneously and independently constructed from flows of correlated powers of Bessel processes by Y. Yu. The second application strengthens an earlier result on convergence of rescaled ``true'' self-avoiding walks to the ``true'' self-repelling motion of B. Tóth and W. Werner, and streamlines its proof. Our results can be thought of as an ``inversion'' of the augmented joint Ray-Knight theorem to get a functional limit theorem for self-interacting random walks.