Chain of matrices, loop equations and topological recursion

Details Chain of matrices, loop equations and topological recursion Chain of matrices, loop equations and topological recursion

2009-11-26

arxiv.org/abs/0911.5089v1

Physics

Mathematical Physics

28 pages, 1 figure, contribution to The Oxford Handbook of Random Matrix Theory

Scientific paper

Random matrices are used in fields as different as the study of multi-orthogonal polynomials or the enumeration of discrete surfaces. Both of them are based on the study of a matrix integral. However, this term can be confusing since the definition of a matrix integral in these two applications is not the same. These two definitions, perturbative and non-perturbative, are discussed in this chapter as well as their relation. The so-called loop equations satisfied by integrals over random matrices coupled in chain is discussed as well as their recursive solution in the perturbative case when the matrices are Hermitean.

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