Current fluctuations of a self-interacting diffusion on a ring

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.