**Astérisque**

Volume: 388;
2017;
201 pp;
Softcover

MSC: Primary 60; 81; 46;
**Print ISBN: 978-2-85629-853-4
Product Code: AST/388**

List Price: $67.00

Individual Member Price: $53.60

# The Master Field on the Plane

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*Thierry Lévy*

A publication of the Société Mathématique de France

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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.