Random matrices and loop equations

Abstract

Abstract This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.