Asymptotics of Brownian motions on classical Lie groups, the master field on the plane, and the Makeenko-Migdal equations.

Thierry Lévy (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 the Wilson loops admit a deterministic limit, towards which they converge at a speed which is uniform on sets of loops with bounded length and for which we obtain a simple explicit upper bound. 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.

Mots clés: Brownian motion on Lie groups; Asymptotic freeness; Brauer algebra; Large N Yang-Mills; Amperean area; Master field; Group of rectifiable loops; Makeenko-Migdal equations

Domaines: Physique mathématique [math-ph]

Thierry Lévy. Asymptotics of Brownian motions on classical Lie groups, the master field on the plane, and the Makeenko-Migdal equations.. 2011. ⟨hal-00650653v1⟩