### 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 $\{X_1,\ldots,X_n\}$ 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 $d \in \{1 ,2\}$.

Translated title of the contribution | Random walk with barycentric self-interaction |
---|---|

Original language | English |

Pages (from-to) | 855 - 888 |

Number of pages | 34 |

Journal | Journal of Statistical Physics |

Volume | 143 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Jun 2011 |

## Fingerprint Dive into the research topics of 'Random walk with barycentric self-interaction'. Together they form a unique fingerprint.

## Cite this

Comets, F., Menshikov, M. V., Volkov, S., & Wade, AR. (2011). Random walk with barycentric self-interaction.

*Journal of Statistical Physics*,*143*(5), 855 - 888. https://doi.org/10.1007/s10955-011-0218-7