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

Abstract : We consider a system consisting of a planar random walk on a square lattice, submitted to stochastic elementary local deformations. Both numerical and theoretical results are reported. Depending on the deformation transition rates, and specifically on a parameter 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: particles of the first one move in a landscape defined by particles of the second one, and vice-versa. This can be viewed as an inhomogeneous exclusion process. 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.
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Submitted on : Tuesday, May 23, 2006 - 7:24:08 PM
Last modification on : Thursday, February 3, 2022 - 11:13:48 AM
Long-term archiving on: : Sunday, April 4, 2010 - 8:54:05 PM

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  • HAL Id : inria-00071977, version 1

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Guy Fayolle, Cyril Furtlehner. Dynamical Windings of Random Walks and Exclusion Models.Part I: Thermodynamic Limit. [Research Report] RR-4608, INRIA. 2002. ⟨inria-00071977⟩

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