  We study planar random holonomy fields which are processes indexed by paths
on the plane which behave well under the concatenation and
orientation-reversing operations on paths. We define the Planar Markovian
Holonomy Fields as planar random holonomy fields which satisfy some
independence and invariance by area-preserving homeomorphisms properties. We
use the theory of braids in the framework of classical probabilities: for
finite and infinite random sequences the notion of invariance by braids is
defined and we prove a new version of the de-Finetti's Theorem. This allows us
to construct a family of Planar Markovian Holonomy Fields, the Yang-Mills
fields, and we prove that any regular Planar Markovian Holonomy Field is a
planar Yang-Mills field. This family of planar Yang-Mills fields can be
partitioned into three categories according to the degree of symmetry: we study
some equivalent conditions in order to classify them. Finally, we recall the
notion of Markovian Holonomy Fields and construct a bridge between the planar
and non-planar theories. Using the results previously proved in the article, we
compute, for any Markovian Holonomy Field, the "law" of any family of
contractible loops drawn on a surface.
