Current fluctuations of a self-interacting diffusion on a ring

TL;DR

This paper studies the fluctuations of current in a self-interacting diffusion on a ring, revealing a phase transition between delocalized and localized states and developing methods to estimate current fluctuations.

Contribution

It introduces a phase diagram for the SID, analyzes current fluctuations with novel approximations, and extends large deviation theory to non-Markovian processes.

Findings

Identifies a delocalisation-localisation phase transition.

Provides bounds and estimates for current fluctuations.

Suggests a phase transition at the onset of localization.

Abstract

We investigate fluctuations in the average speed or current of a self-interacting diffusion (SID) on a ring, mimicking the non-Markovian behaviour of an agent influenced by its own path. We derive the SID's phase diagram, showing a delocalisation-localisation phase transition from self-repelling to self-attracting. Current fluctuations are analysed using: (i) an adiabatic approximation, where the system reaches its stationary distribution before developing current fluctuations, and (ii) an original extension of level 2.5 large deviations for Markov processes combined with perturbation theory. Both methods provide lower bounds to current fluctuations, with the former tighter than the latter in all localised regimes, and both equally tight in the self-repelling region. Both methods accurately estimate the asymptotic variance and suggests a phase transition at the onset of the localised regime.