Large N expansions in matrix models
Monday, June 25, 2012 - 9:00am - 10:30am
Large dimension expansion of matrix integrals has long been used to study combinatorial objects such as maps, that is graphs sorted by the genus of the surfaces in which they can be properly embedded. In this course, we shall study these expansions. We will first motivate this approach and consider formal expansions. The asymptotics expansions will require a more detailed study of the properties of the spectral measure of random matrices. We shall first consider classical one-matrix models and study the asymptotics of their spectral measure; law of large numbers, central limit theorem, concentration inequalities and large deviations properties. In a second time we shall obtain a full asymptotic expansion of the Cauchy-Stieljes transform of the spectral measure, hence provided a rigorous derivation of the asymptotic topological expansion of matrix integrals in the one matrix case. In the second part of the course, we shall tackle several matrix models, and their application in the enumeration of colored maps and sophisticated combinatorial objects. If time allows, we shall effectively use random matrices to count combinatorial models such as planar maps or loop models.