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.