13 - Matrix models

Summary

Matrix models first appeared in statistical mechanics and nuclear physics [Wig51, Dys62] and turned out to be very useful in the analysis of various physical systems where the energy levels of a complicated Hamiltonian can be approximated by the distribution of eigenvalues of a random matrix. The statistical averaging is then replaced by averaging over an appropriate ensemble of random matrices. This idea has been applied, in particular, in studying the low-energy chiral properties of QCD [SV93, VZ93].

Matrix models possess some features of multicolor QCD described in Chapter 11 but are simpler and can often be solved as N → ∞ (i.e. in the planar limit) using the methods proposed for multicolor QCD. For the simplest case of the Hermitian one-matrix model, the genus expansion in 1/N can be constructed.

The Hermitian one-matrix model is related to the problem of enumeration of graphs. Its explicit solution at large N was first obtained by Brézin, Itzykson, Parisi and Zuber [BIP78] and inspired a lot of activity in this subject. Further results in this direction are linked to the method of orthogonal polynomials [Bes79, IZ80, BIZ80].

A very interesting application of the matrix models along this line is for the problem of discretization of random surfaces and two-dimensional quantum gravity [Kaz85, Dav85, ADF85, KKM85]. The continuum limits of these matrix models are associated with lower-dimensional conformal field theories and exhibit properties of integrable systems.