Analytic calculations of observables in the non-Abelian lattice gauge theories are available only in the strong-coupling regime g2 → ∞, while one needs g2 → 0 for the continuum limit. When g2 is decreased, the lattice systems can undergo phase transitions as often happens in statistical mechanics.

To look for phase transitions, the mean-field method was first applied to lattice gauge theories [Wil74, BDI74]. It turned out to be useful for studying the first-order phase transitions which very often happen in lattice gauge systems but do not affect the continuum limit.

The second-order phase transitions are better described by the lattice renormalization group method. The approximate Migdal–Kadanoff recursion relations [Mig75, Kad76] were the first implementation of the renormalization group transformation on a lattice, which indicated the absence of a second-order phase transition in the non-Abelian lattice gauge theories and, therefore, quark confinement.

A very powerful method for practical nonperturbative calculations of observables in lattice gauge theories is the numerical Monte Carlo method. This method simulates statistical processes in a lattice gauge system and for this reason is often called a numerical simulation. The idea of applying it to lattice gauge theories is due to Wilson [Wil77], while the practical implementation was done by Creutz, Jacobs and Rebbi [CJR79] for Abelian gauge groups and by Creutz [Cre79, Cre80] for the SU (2) and SU (3) groups.

The simplest models, which become solvable in the limit of a large number of field components, deal with a field which has N components forming an O(N) vector in an internal symmetry space. A model of this kind was first considered by Stanley [Sta68] in statistical mechanics and is known as the spherical model. The extension to quantum field theory was made by Wilson [Wil73] both for the four-Fermi and ϕ4 theories.

Within the framework of perturbation theory, the four-Fermi interaction is renormalizable only in d = 2 dimensions and is nonrenormalizable for d > 2. The 1/N-expansion resums perturbation-theory diagrams after which the four-Fermi interaction becomes renormalizable to each order in 1/N for 2 ≤ d < 4. An analogous expansion exists for the nonlinear O(N) sigma model. The ϕ4 theory remains “trivial” in d = 4 to each order of the 1/N-expansion and has a nontrivial infrared-stable fixed point for 2 < d < 4.

The 1/N-expansion of the vector models is associated with a resummation of Feynman diagrams. A very simple class of diagrams – the bubble graphs – survives to the leading order in 1/N. This is why the large-N limit of the vector models is solvable. Alternatively, the large-N solution is nothing but a saddle-point solution in the path-integral approach. The existence of the saddle point is a result of the fact that N is large.

The large-N reduction was first discovered in 1982 by Eguchi and Kawai [EK82], who showed that the SU(N) Yang–Mills theory on a d-dimensional space-time is equivalent at N = ∞ to the one at a point. This construction is based on an extra symmetry of the reduced model which should not be broken spontaneously.

Soon after that it was recognized that this symmetry is, in fact, broken for d > 2. Two ways were proposed to cure the construction: the quenching prescription [BHN82] and the twisting prescription [GO83a]. Each of these two prescriptions results in a reduced model which recovers multicolor QCD both on the lattice and in the continuum.

While the reduced models look like a great simplification, since the space-time is reduced to a point, they still involve an integration over d infinite matrices which is, in fact, a continual path integral. For some years it was not clear whether or not this is a real simplification of the original theory which can make it solvable, so the point of view on the reduced models was that they are just an elegant representation at large N.

The recent interest in reduced models has arisen from the matrix-model formulation [BFS97, IKK97] of M-theory combining all types of superstring theories. The novel point of view on the reduced models is that they are equivalent [CDS98] to gauge theories on noncommutative space.