Random walk with barycentric self-interaction

Abstract

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (Xn)n∈ℕ (ℕ:={1,2,3,…}) which is repelled or attracted by the centre of mass of its previous trajectory. The walk’s trajectory (X1,…,Xn) models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean drift TeX for ρ∈ℝ and β≥0. When β<1 and ρ>0, we show that Xn is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n−1/(1+β)Xn converges almost surely to some random vector. When β∈(0,1) there is sub-ballistic rate of escape. When β≥0 and ρ∈ℝ we give almost-sure bounds on the norms ‖Xn‖, which in the context of the polymer model reveal extended and collapsed phases. Analysis of the random walk, and in particular of Xn−Gn, leads to the study of real-valued time-inhomogeneous non-Markov processes (Zn)n∈ℕ on [0,∞) with mean drifts of the form 0.1TeX where β≥0 and ρ∈ℝ. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on ℤd from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Zn satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β≥0) of Xn−Gn for our self-interacting walk.