  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.
