### Abstract

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 - Jun 2011 |

### Fingerprint

### Keywords

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

### Cite this

*Journal of Statistical Physics*,

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

}

*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.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Random walk with barycentric self-interaction

AU - Comets, Francis

AU - Menshikov, Mikhail V.

AU - Volkov, Stanislav

AU - Wade, Andrew R.

PY - 2011/6

Y1 - 2011/6

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

UR - http://www.scopus.com/inward/record.url?scp=79958739191&partnerID=8YFLogxK

U2 - 10.1007/s10955-011-0218-7

DO - 10.1007/s10955-011-0218-7

M3 - Article

VL - 143

SP - 855

EP - 888

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5

ER -