We consider a system consisting of a planar random walk on a square lattice,submitted to stochastic elementary local deformations. Depending on thedeformation transition rates, and specifically on a parameter $\eta$ whichbreaks the symmetry between the left and right orientation, the windingdistribution of the walk is modified, and the system can be in three differentphases: folded, stretched and glassy. An explicit mapping is found, leading toconsider the system as a coupling of two exclusion processes. For all closed orperiodic initial sample paths, a convenient scaling permits to show aconvergence in law (or almost surely on a modified probability space) to acontinuous curve, the equation of which is given by a system of two non linearstochastic differential equations. The deterministic part of this system isexplicitly analyzed via elliptic functions. In a similar way, by using a formalfluid limit approach, the dynamics of the system is shown to be equivalent to asystem of two coupled Burgers' equations.
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机译：我们考虑一个系统，该系统由在正方形晶格上的平面随机游动组成，并服从随机基本局部变形。取决于变形转变率，特别是取决于破坏左右方向对称性的参数$ \ eta $，可以修改步道的缠绕分布，并且系统可以处于三个不同的阶段：折叠，拉伸和玻璃状。找到显式映射，从而将系统视为两个排除过程的耦合。对于所有闭合的非周期初始样本路径，方便的缩放允许显示规律上的收敛（或几乎肯定在修改的概率空间上）到连续曲线，该方程的方程由两个非线性随机微分方程组给出。通过椭圆函数明确地分析了该系统的确定性部分。以类似的方式，通过使用形式流体极限方法，系统的动力学表现为等效于两个耦合Burgers方程的系统。
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