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In theoretical physics, one often considers gauge theory models whose symmetries are groups of large matrices, notably in the special unitary group $SU(N)$. The limit of such theories for $N\to \infty$ (“large number of colours-limit”) has often remarkable properties and string-like features.
This limit is sometimes appearing in its own right, but sometimes it is just considered as an approximation for a system with fixed finite $N$. One of the features is that in the large N limit is that non-planar Feynman diagrams lose their importance and that the correlation functions satisfy certain decoupling/factorization rule. The behaviour is studied in terms of expansion in $1/N$ whose square has a simialr role to Planck constant in semiclassical approximation limit of quantum mechanics.
On the large N limit of QCD at fixed 't Hooft coupling in terms of planar Feynman diagrams:
Gerard 't Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B72 (1974) 461 (spire:80491, doi:10.1016/0550-3213(74)90154-0)
Gerard 't Hooft, Large N, workshop lecture (hep-th/0204069)
See also:
A. A. Migdal, Loop equations and 1/N expansion, Physics Reports, 102 (4), 199-290 (1983) doi
Laurence G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982) pdf
Sidney Coleman, 1/N, in Aspects of Symmetry, Cambridge University Press 1985.
A. V. Manohar, Large N QCD, L:es Houches Lecture 2004, pdf
A. Jevicki, Instantons and the $1/N$ expansion in nonlinear $\sigma$ models, Phys. Rev. D 20, 3331–3335 (1979) pdf
Juan Maldacena, The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2:231-252, 1998 hep-th/9711200; Wilson loops in large N field theories, hep-th/9803002.
E. Brézin, S.R. Wadia, eds. The Large N Expansion in Quantum Field Theory and Statistical Physics, a book collection of reprinted historical articles, gBooks
Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks
M. Bershadsky, Z. Kakushadze, C. Vafa, String expansion as large N expansion of gauge theories, Nucl.Phys. B523 (1998) 59-72hep-th/9803076, doi
G.T. Horowitz, H. Ooguri, Spectrum of large N gauge theory from supergravity, hep-th/9802116
Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, arxiv/1309.7333
wikipedia 1/N expansion
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