Random walk with barycentric self-interaction

Francis Comets, Mikhail V. Menshikov, Stanislav Volkov, and Andrew R. Wade

Journal of Statistical Physics, 143, no. 5, June 2011, 855–888. DOI: 10.1007/s10955-011-0218-7

Abstract

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 $(X_1,\ldots,X_n)$ 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 $\rho \in \mathbb{R}$ 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 $(Z_n ; n \geq 1)$ on $[0,\infty)$ with mean drifts at $x$ of the form $\rho x^{-\beta} - (x/n)$, where $\beta \geq 0$ 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 $Z_n$ 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.

Further remarks

The first picture shows a simulation of a trajectory of the random walk (red) and its barycentre (blue) for a choice of parameters for which the interaction is repulsive and for which in the paper we prove that the walk satisfies a super-diffusive but sub-ballistic strong law of large numbers, and has a limiting direction. The second picture shows a simple symmetric random walk and its centre of mass process. The random walk is recurrent, and (by a result of Grill [MR0933296]) the centre of mass is transient. An application of the results in our paper shows that the displacement between the walk and the current centre of mass is recurrent. For a generalization of Grill's work, see this subsequent paper.