Diffusive limit for self-repelling Brownian polymers in three and more dimensions

TL;DR

This paper proves that in three or more dimensions, the self-repelling Brownian polymer exhibits diffusive behavior with its displacement converging to a Wiener process, confirming long-standing conjectures.

Contribution

It extends 1D results to higher dimensions, establishing diffusive scaling and convergence to Brownian motion for SRBP in non-recurrent dimensions.

Findings

Displacement scales diffusively in 3+ dimensions

Finite-dimensional distributions converge to Wiener process

Identifies stationary and ergodic environment distribution

Abstract

The self-repelling Brownian polymer model (SRBP) initiated by Durrett and Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)]. In both cases, a random motion in space is pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of SRBP in the non-recurrent dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko (2009)], we identify a natural stationary (in time) and ergodic distribution of the environment (essentially, smeared-out occupation time measure of the process), as seen from the moving particle. As main result we prove that in three and more dimensions, in this stationary (and ergodic) regime, the displacement of the moving particle scales diffusively and its finite dimensional distributions converge to those of a Wiener process. This result settles part of the conjectures (based on non-rigorous renormalization group arguments) in [Amit-Parisi-Peliti (1983)]. The main tool is the non-reversible version of the Kipnis--Varadhan-type CLT for additive functionals of ergodic Markov processes and the graded sector condition of [Sethuraman-Varadhan-Yau (2000)].