Random walk with barycentric self-interaction

We study the asymptotic behaviour of a $d$-dimensional self-interacting random walk $X_n$ 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 $\{X_1,\ldots,X_n\}$ can be viewed as a model for a random polymer chain in either poor or good solvent. Analysis of the random walk, and in particular $X_n - G_n$, leads to the study of time-inhomogeneous non-Markov processes $(Z_n)_{n \in \N}$ on $[0,\infty)$ with one-step mean drifts of the form \begin{equation} \label{star} \Exp [ Z_{n+1} - Z_n \mid Z_n = x ] \approx \rho x^{-\beta} - \frac{x}{n}, \end{equation} where $\beta > 0$ 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. We give a recurrence classification for processes $Z_n$ satisfying (\ref{star}), which enables us to deduce asymptotic properties of $X_n - G_n$ for our self-interacting random walk. We also give almost-sure bounds on $\|X_n\|$, which in view of the interpretation of $\{X_1,\ldots,X_n\}$ as a model for a polymer chain reveal four distinct phases, including extended, diffusive, and collapsed. Our results yield the following apparently new observation about simple symmetric random walk on $\Z^d$: the distance between the position of the walk at time $n$ and the centre of mass of its first $n$ positions lies in some finite interval $[0,b]$ for infinitely many $n$ with probability 1 if and only if $d \in \{1 ,2\}$.