Dynamical Windings of Random Walks and Exclusion Models. Part I: Thermodynamic Limit

We consider a system consisting of a planar random walk on a square lattice, submitted to stochastic elementary local deformations. Depending on the deformation transition rates, and specifically on a parameter \(\eta\) which breaks the symmetry between the left and right orientation, the winding di...

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Bibliographic Details
Published in:arXiv.org 2002-11
Main Authors: Fayolle, Guy, Furtlehner, Cyril
Format: Article
Language:English
Subjects:
Coils (windings)
Deformation
Differential equations
Elliptic functions
Mapping
Random walk
Random walk theory
Online Access:Get full text
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Summary:We consider a system consisting of a planar random walk on a square lattice, submitted to stochastic elementary local deformations. Depending on the deformation transition rates, and specifically on a parameter \(\eta\) which breaks the symmetry between the left and right orientation, the winding distribution of the walk is modified, and the system can be in three different phases: folded, stretched and glassy. An explicit mapping is found, leading to consider the system as a coupling of two exclusion processes. For all closed or periodic initial sample paths, a convenient scaling permits to show a convergence in law (or almost surely on a modified probability space) to a continuous curve, the equation of which is given by a system of two non linear stochastic differential equations. The deterministic part of this system is explicitly analyzed via elliptic functions. In a similar way, by using a formal fluid limit approach, the dynamics of the system is shown to be equivalent to a system of two coupled Burgers' equations.
ISSN:2331-8422
DOI:10.48550/arxiv.0211141