  This study in centered on models accounting for stochastic deformations of
sample paths of random walks, embedded either in $\mathbb{Z}^2$ or in
$\mathbb{Z}^3$. These models are immersed in multi-type particle systems with
exclusion. Starting from examples, we give necessary and sufficient conditions
for the underlying Markov processes to be reversible, in which case their
invariant measure has a Gibbs form. Letting the size of the sample path
increase, we find the convenient scalings bringing to light phase transition
phenomena. Stable and metastable configurations are bound to time-periods of
limiting deterministic trajectories which are solution of nonlinear
differential systems: in the example of the ABC model, a system of
Lotka-Volterra class is obtained, and the periods involve elliptic,
hyper-elliptic or more general functions. Lastly, we discuss briefly the
contour of a general approach allowing to tackle the transient regime via
differential equations of Burgers' type.
