The master field on the plane

TL;DR

This paper investigates the large N behavior of Brownian motions on classical groups, extends convergence results, and analyzes the large N limit of the Yang-Mills measure on the plane, including Wilson loop expectations and Schwinger-Dyson equations.

Contribution

It extends non-commutative distribution convergence to orthogonal and symplectic groups, constructs the large N Yang-Mills measure, and rigorously derives the Makeenko-Migdal equations.

Findings

Convergence of Wilson loops to deterministic limits

Explicit speed estimates for convergence of expectations

Derivation of recursive computation via differential systems

Abstract

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.