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

TL;DR

This paper studies a planar random walk with local deformations, revealing phase transitions and mapping the system to coupled exclusion processes, with results on convergence to stochastic differential equations and coupled Burgers' equations.

Contribution

It introduces a novel analysis of windings in random walks with local deformations, connecting them to exclusion processes and deriving limiting stochastic differential equations.

Findings

Identification of three phases: folded, stretched, and glassy.

Explicit mapping to coupled exclusion processes.

Convergence to a system of nonlinear stochastic differential equations.

Abstract

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.