Random walk with barycentric selfinteraction
Comets, Francis and Menshikov, Mikhail V. and Volkov, Stanislav and Wade, Andrew R. (2011) Random walk with barycentric selfinteraction. Journal of Statistical Physics, 143 (5). pp. 855888. ISSN 00224715
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Abstract
We study the asymptotic behaviour of a $d$dimensional selfinteracting 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 onestep 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 superdiffusive law of large numbers: $n^{1/(1+\beta)} X_n$ converges almost surely to some random vector. When $\beta \in (0,1)$ there is subballistic rate of escape. For $\beta \geq 0$ we give almostsure 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 realvalued timeinhomogeneous nonMarkov 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 timedependent 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 selfinteracting walk.


Item type: Article ID code: 35958 Dates: DateEventJune 2011PublishedKeywords: limiting direction , law of large numbers, selfavoiding walk, selfinteracting 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: 02 Oct 2021 01:32 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/35958