# Random walk with barycentric self-interaction

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

Research output: Contribution to journalArticle

5 Citations (Scopus)

### 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.
Original language English 855-888 34 Journal of Statistical Physics 143 5 https://doi.org/10.1007/s10955-011-0218-7 Published - Jun 2011

### Fingerprint

Centrobaric
Barycentre
random walk
Walk
center of mass
Random walk
Recurrence
Polymers
Interaction
trajectories
Trajectory
Simple Random Walk
Law of large numbers
polymers
interactions
Ballistics
Asymptotic Theory
Random Vector
Regularity Conditions
norms

### Keywords

• limiting direction
• law of large numbers
• self-avoiding walk
• self-interacting random walk
• random polymer

### Cite this

Comets, F., Menshikov, M. V., Volkov, S., & Wade, A. R. (2011). Random walk with barycentric self-interaction. Journal of Statistical Physics, 143(5), 855-888. https://doi.org/10.1007/s10955-011-0218-7
Comets, Francis ; Menshikov, Mikhail V. ; Volkov, Stanislav ; Wade, Andrew R. / Random walk with barycentric self-interaction. In: Journal of Statistical Physics. 2011 ; Vol. 143, No. 5. pp. 855-888.
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Comets, F, Menshikov, MV, Volkov, S & Wade, AR 2011, 'Random walk with barycentric self-interaction', Journal of Statistical Physics, vol. 143, no. 5, pp. 855-888. https://doi.org/10.1007/s10955-011-0218-7

Random walk with barycentric self-interaction. / Comets, Francis; Menshikov, Mikhail V.; Volkov, Stanislav; Wade, Andrew R.

In: Journal of Statistical Physics, Vol. 143, No. 5, 06.2011, p. 855-888.

Research output: Contribution to journalArticle

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T1 - Random walk with barycentric self-interaction

AU - Comets, Francis

AU - Menshikov, Mikhail V.

AU - Volkov, Stanislav

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N2 - 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.

AB - 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.

KW - limiting direction

KW - law of large numbers

KW - self-avoiding walk

KW - self-interacting random walk

KW - random polymer

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JO - Journal of Statistical Physics

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