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

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

2002-11-07

arxiv.org/abs/cond-mat/0211141v2

J. Stat. Phys. Vol. 114, No 1-2 (2004), 229-260

Physics

Condensed Matter

Statistical Mechanics

31 pages, 13 figures. Pages 5,6,8,9,10,12,23 color printed. INRIA Report 4608

Scientific paper

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.

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