  We study the large N asymptotics of the Brownian motions on the orthogonal,
unitary and symplectic groups, extend the convergence in non-commutative
distribution originally obtained by Biane for the unitary Brownian motion to
the orthogonal and symplectic cases, and derive explicit estimates for the
speed of convergence in non-commutative distribution of arbitrary words in
independent Brownian motions. Using these results, we construct and study the
large N limit of the Yang-Mills measure on the Euclidean plane with orthogonal,
unitary and symplectic structure groups. We prove that each Wilson loop
converges in probability towards a deterministic limit, and that its
expectation converges to the same limit at a speed which is controlled
explicitly by the length of the loop. In the course of this study, we reprove
and mildly generalise a result of Hambly and Lyons on the set of tree-like
rectifiable paths. Finally, we establish rigorously, both for finite N and in
the large N limit, the Schwinger-Dyson equations for the expectations of Wilson
loops, which in this context are called the Makeenko-Migdal equations. We study
how these equations allow one to compute recursively the expectation of a
Wilson loop as a component of the solution of a differential system with
respect to the areas of the faces delimited by the loop.
