Random Walk with Barycentric Self-interaction

Abstract

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X _n ) _n∈ℕ (ℕ:={1,2,3,…}) which is repelled or attracted by the centre of mass $G_{n} = n^{-1} \sum_{i=1}^{n} X_{i}$ of its previous trajectory. The walk’s trajectory (X ₁,…,X _n ) 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

$$\mathbb{E}[X_{n+1} - X_n \mid X_n - G_n = \mathbf{x}] \approx\rho\|\mathbf{x}\|^{-\beta}\hat{ \mathbf{x}}$$

for ρ∈ℝ and β≥0. When β<1 and ρ>0, we show that X _n is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n ^−1/(1+β) X _n 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 ‖X _n ‖, which in the context of the polymer model reveal extended and collapsed phases.

Analysis of the random walk, and in particular of X _n −G _n , leads to the study of real-valued time-inhomogeneous non-Markov processes (Z _n ) _n∈ℕ on [0,∞) with mean drifts of the form

$$ \mathbb{E}[ Z_{n+1} - Z_n \mid Z_n = x ] \approx\rho x^{-\beta} - \frac {x}{n},$$

(0.1)

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 ℤ^d from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Z _n satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β≥0) of X _n −G _n for our self-interacting walk.