# Random walk with barycentric self-interaction

Comets, Francis and Menshikov, Mikhail V. and Volkov, Stanislav and Wade, Andrew R.
(2011)
*Random walk with barycentric self-interaction.*
Journal of Statistical Physics, 143 (5).
pp. 855-888.
ISSN 0022-4715

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## Abstract

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.

Item type: | Article |
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ID code: | 35958 |

Keywords: | limiting direction , law of large numbers, self-avoiding walk, self-interacting random walk, random polymer , Physics, Mathematical Physics, Statistical and Nonlinear Physics |

Subjects: | Science > Physics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Pure Administrator |

Date deposited: | 17 Nov 2011 09:57 |

Last modified: | 24 Jan 2019 06:24 |

Related URLs: | |

URI: | https://strathprints.strath.ac.uk/id/eprint/35958 |

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