The master field on the plane.

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 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.

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

Domaines: Physique mathématique [math-ph]

Thierry Lévy. The master field on the plane.. Asterisque, 2017, 388. ⟨hal-00650653v2⟩