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The master field on the plane
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...
Enregistré dans:
| Auteur principal : | |
|---|---|
| Format : | Livre |
| Langue : | anglais |
| Titre complet : | The master field on the plane / Thierry Lévy |
| Publié : |
Paris :
Société mathématique de France
, DL 2017 |
| Description matérielle : | 1 vol. (IX-201 p.) |
| Collection : | Astérisque ; 388 |
| Sujets : | |
| Documents associés : | Autre format:
The master field on the plane Fait partie de l'ensemble: Astérisque |
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| 200 | 1 | |a The master field on the plane |f Thierry Lévy | |
| 210 | |a Paris |c Société mathématique de France |d DL 2017 | ||
| 215 | |a 1 vol. (IX-201 p.) |c ill. |d 25 cm | ||
| 305 | |a N° de : "Astérisque", ISSN 0303-1179, (2017) n° 388 | ||
| 320 | |a Bibliogr. p. [193]-197. Index | ||
| 330 | |a 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 fulfil part of a progam outlined by Singer by contructing and studying 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 wich 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. | ||
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