Large-N limit as a classical limit: Baryon in two-dimensional QCD and multi-matrix models

In this thesis, I study the limit of a large number of colors ( N) in a non-abelian gauge theory. It corresponds to a classical limit where fluctuations in gauge-invariant observables vanish. The large-dimension limit for rotation-invariant variables in atomic physics is given as an example of a classical limit for vector models.;The baryon is studied in Rajeev's reformulation of two-dimensional QCD in the large-N limit: a bilocal classical field theory for color-singlet quark bilinears, whose phase space is an infinite grassmannian. In this approach, 't Hooft's integral equation for mesons describes small oscillations around the vacuum. Baryons are topological solitons on a disconnected phase space, labelled by baryon number. The form factor of the ground-state baryon is determined variationally on a succession of increasing-rank submanifolds of the phase space. These reduced dynamical systems are rewritten as interacting parton models, allowing us to reconcile the soliton and parton pictures. The rank-one ansatz leads to a Hartree-type approximation for colorless valence quasiparticles, which provides a relativistic two-dimensional realization of Witten's ideas on baryon structure in the 1/N expansion. The antiquark content of the baryon is small and vanishes in the chiral limit. The valence-quark distribution is used to model parton distribution functions measured in deep inelastic scattering. A geometric adaptation of steepest descent to the grassmannian phase space is also given.;Euclidean large-N multi-matrix models are reformulated as classical systems for U(N) invariants. The configuration space of gluon correlations is a space of non-commutative probability distributions. Classical equations of motion (factorized loop equations) contain an anomaly that leads to a cohomological obstruction to finding an action principle. This is circumvented by expressing the configuration space as a coset space of the automorphism group of the tensor algebra. The action principle is interpreted as the partial Legendre transform of the entropy of operator-valued random variables. The free energy and correlations in the N → infinity limit are determined variationally. The simplest variational ansatz is an analogue of mean-field theory. The latter compares well with exact solutions and Monte-Carlo simulations of one and two-matrix models away from phase transitions.