Random walk with barycentric self-interaction
Résumé
We study the asymptotic behaviour of a $d$-dimensional self-interacting random walk $(X_n)_{n \in \N}$ ($\N := \{1, 2, 3, ...\}$) 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 of order $\rho \| X_n - G_n \|^{-\beta}$ in the $X_n - G_n$ direction, where $\rho \in \R$ and $\beta \geq 0$. When $\beta <1$ and $\rho>0$, we show that $X_n$ is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: $n^{-1/(1+\beta)} X_n$ converges almost surely to some random vector. When $\beta <1$ there is sub-ballistic rate of escape. We also 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 $(Z_n)_{n \in \N}$ on $[0,\infty)$ with mean drifts at $x$ of the form $\rho x^{-\beta} - (x/n)$, 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 $Z_n$ of the type 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.