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Astérisque - Titles - 388 (2017)

Titles < 2017

The master field on the plane
Thierry Lévy
Astérisque 388 (2017), x+201 pages
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Résumé :
Le champ maître sur le plan
Nous étudions le comportement asymptotique des mouvements browniens sur les groupes orthogonaux, unitaires et symplectiques lorsque le rang de ces groupes tend vers l'infini. Nous étendons aux cas orthogonal et symplectique le résultat de convergence en distribution non-commutative initialement obtenu par Biane pour le mouvement brownien unitaire et nous établissons une estimation explicite et simple de la vitesse de convergence en distribution non-commutative pour un mot arbitraire en des accroissements indépendants de mouvements browniens. Ces résultats nous permettent de construire et d'étudier la limite de la mesure de Yang-Mills sur le plan euclidien avec des groupes de structure orthogonaux, unitaires et symplectiques, lorsque le rang de ces groupes tend vers l'infini. Ce faisant, nous réalisons une partie d'un programme décrit par Singer. Nous prouvons que chaque boucle de Wilson converge en probabilité vers une limite déterministe et que son espérance converge vers la même limite à une vitesse qui est explicitement contrôlée par la longueur du lacet. Au cours de cette étude, nous redémontrons et généralisons marginalement un résultat de Hambly et Lyons concernant l'ensemble des chemins rectifiables arborescents (ceux que les auteurs appellent tree-like). Enfin, nous démontrons rigoureusement, en rang fini et à la limite où le rang tend vers l'infini, les équations de Schwinger-Dyson pour les espérances des boucles de Wilson, qui dans ce contexte s'appellent les équations de Makeenko-Migdal. Nous examinons la manière dont ces équations permettent de calculer récursivement les espérances des boucles de Wilson, chacune étant une composante de la solution d'un système différentiel par rapport aux aires des faces délimitées par le lacet.

Mots-clefs : Mouvement brownien sur les groupes de Lie, liberté asymptotique, algèbre de Brauer, limite de la théorie de Yang-Mills lorsque le rang tend vers l'infini, aire ampéréenne, champ maître, groupe des lacets rectifiables, équations de Makeenko-Migdal

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 increments of Brownian motions. Using these results, we fulfil 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. 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.

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

Class. math. : 60B20, 81T13, 46L54


ISBN : 978-2-85629-853-4
ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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