The author studies the large \(N\) asymptotics of the Brownian motions on the orthogonal, unitary and symplectic groups, extends the convergence in non-commutative distribution originally obtained by Biane for the unitary Brownian motion to the orthogonal and symplectic cases, and derives explicit estimates for the speed of convergence in non-commutative distribution of arbitrary words in independent increments of Brownian motions.

Using these results, the author fulfills part of a program outlined by Singer by constructing and studying the large \(N\) limit of the Yang–Mills measure on the Euclidean plane with orthogonal, unitary, and symplectic structure groups. He proves 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, the author reproves and mildly generalizes a result of Hambly and Lyons on the set of tree-like rectifiable paths.

Finally, the author rigorously establishes, 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. The author studies 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.