The large-N expansion of unitary-matrix models
  The general features of the 1/N expansion in statistical mechanics and quantum field theory are briefly reviewed both from the theoretical and from the phenomenological point of view as an introduction to a more detailed analysis of the large-N properties of spin and gauge models possessing the symmetry group $SU(N) \times SU(N)$.   An extensive discussion of the known properties of the single-link integral (equivalent to YM_2 and one-dimensional chiral models) includes finite-$N$ results, the external field solution, properties of the determinant, and the double scaling limit.   Two major classes of solvable generalizations are introduced: one-dimensional closed chiral chains and models defined on a $d-1$ dimensional simplex. In both cases large-N solutions are presented with emphasis on their double scaling properties.   The available techniques and results concerning unitary-matrix models that correspond to asymptotically free quantum field theories (two-dimensional chiral models and four-dimensional QCD) are discussed, including strong-coupling methods, reduced formulations, and the Monte Carlo approach. 
Paolo Rossi, Massimo Campostrini, and Ettore Vicari