Random walk with barycentric self-interaction

Abstract

We study the asymptotic behaviour of a $d$-dimensional self-interacting random walk $X_n$ ($n=1,2,...$) 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 directed either towards or away from its current centre of mass $G_n$ and of magnitude $\Vert X_n-G_n{\Vert }^{-\beta }$ for $\beta \ge 0$. When $\beta <1$ and the radial drift is outwards, we show that $X_n$ is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: $n^{-1/(1+\beta )}{X}_n$ converges almost surely to some random vector. When $\beta \in (0,1)$ there is sub-ballistic rate of escape. For $\beta \ge 0$ we give almost-sure bounds on the norms $\Vert X_n\Vert$, 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 $Z_n$ on $[0,\infty )$ with mean drifts at $x$ given approximately by $\rho x^{-\beta }-(x/n)$, where $\beta \ge 0$ and $\rho \in \mathbb{R}$. 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 ${\mathbb{Z}}^d$ from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes $Z_n$ just described, which enables us to deduce the complete recurrence classification (for any $\beta \ge 0$) of $X_n-G_n$ for our self-interacting walk.

36 pages, 2 colour figures; v2: minor revision, some corrections

References in corpus (3)

• A survey of random processes with reinforcement
• An asymptotic result for Brownian polymers
• Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains