Exact Results in Quantum Theory

Random matrices first arose in the works of Wishart in the late 1920s. Later, in the mid-1950s, Wigner introduced them as a tool for modelling spectra of heavy nuclei. Since then, these objects have appeared in many other branches of mathematics and physics and have been observed to have many more applications, such as to combinatorial problems of counting various types of graphs. One may consider various types of random matrices, the simplest model having a single random matrix satisfying certain symmetry conditions such as being real, symmetric or hermitian. The study of statistical properties of such random matrix ensembles boils down to the analysis of certain N-fold integrals, N being the size of the square matrix. In fact, these integrals belong to the class of so-called ß-ensemble integrals, which can be thought of as representing the partition function of a one-dimensional Coulomb gas subject to an external potential V. Indeed, at specific values of the parameter ß, to wit, ß=1,2,4, one recovers the non-trivial part of partition functions associated with the so-called matrix model subordinate to the symmetric (the orthogonal ensemble), hermitian (the unitary ensemble) and hermitian self-dual (the symplectic ensemble) random matrices. Most of the information of interest to matrix models (and ß-ensembles) is related to the large-N behaviour. Therefore, its extraction demands analysing the N ? behaviour of N-fold integrals. Introducing modern methods allowing one to carry out this sort of asymptotic analysis is the very purpose of these lectures. I shall commence by introducing the ß-ensembles and discussing their relation, for ß=1,2,4, to the aforementioned matrix ensembles. I shall then make a few heuristic connections between partition functions of matrix models and enumeration problems of graphs drawn on surfaces with prescribed genera. Next, I shall proceed by entering deeper into the subject; namely, by introducing the various concepts and tools allowing one to set a convenient framework for carrying out the large-N asymptotic analysis. I shall begin by discussing certain general topological properties of spaces of probability measures over Polish spaces. This will enable me to introduce the concept of large deviations for sequences of probability measures as well as to prove various crucial theorems relevant to these constructions. At that very point, I shall be able to explain how the framework can be utilised as a powerful tool for computing the leading large-N asymptotic behaviour of the ß-ensemble integrals. Time permitting, I shall briefly expand upon techniques allowing one to compute subdominant large-N asymptotics.