Author : Thierry Lévy
Publisher :
ISBN 13 : 9782856298534
Total Pages : 201 pages
Book Rating : 4.2/5 (985 download)
Book Synopsis The Master Field on the Plane by : Thierry Lévy
Download or read book The Master Field on the Plane written by Thierry Lévy and published by . This book was released on 2017 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
