In many physical problems, especially when fluctuations of scales of different orders of magnitude are essential, there is no small parameter which could simplify a study. A typical example is QCD where the effective coupling, describing strong interaction at a given distance, becomes large at large distances so that the interaction really becomes strong.

't Hooft [Hoo74a] proposed in 1974 to use the dimensionality of the gauge group SU(N) as such a parameter, considering the number of colors, N, as a large number and performing an expansion in 1/N. The motivation was an expansion in the inverse number of field components N in statistical mechanics where it is known as the 1/N-expansion, and is a standard method for nonperturbative investigations.

The expansion of QCD in the inverse number of colors rearranges diagrams of perturbation theory in a way which is consistent with a string picture of strong interaction, the phenomenological consequences of which agree with experiment. The accuracy of the leading-order term, which is often called multicolor QCD or large-N QCD, is expected to be of the order of the ratios of meson widths to their masses, i.e. about 10–15%.

While QCD is simplified in the large-N limit, it is still not yet solved. Generically, it is a problem of infinite matrices, rather than of infinite vectors as in the theory of second-order phase transitions in statistical mechanics.