  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 $\| X_n - G_n \|^{-\beta}$ for $\beta
\geq 0$. 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 $\beta \geq 0$ 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 $Z_n$ on $[0,\infty)$ with
mean drifts at $x$ given approximately by $\rho x^{-\beta} - (x/n)$, where
$\beta \geq 0$ and $\rho \in \R$. 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 $Z_n$ just
described, which enables us to deduce the complete recurrence classification
(for any $\beta \geq 0$) of $X_n - G_n$ for our self-interacting walk.
