The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

We study the spectral measure of Gaussian Wigner's matrices and prove that it satis®es a large deviation principle. We show that the good rate function which governs this principle achieves its minimum value at Wigner's semicircular law, which entails the convergence of the spectral measure to the semicircular law. As a conclusion, we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about Voiculescu's non commutative entropy.

We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the "topological recursion" derived in [CE06]. Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following [dMPS95, Joh98, KS10] or for strictly convex potentials by using concentration of measure [AGZ10, Section 2.3]. Doing so, we extend the strategy of [GMS07], from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in [Joh98] and more recently in [KS10]. Here, by taking similar hypotheses, we extend the result to all orders in 1/N .