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We prove the existence of a 1/N expansion in unitary multimatrix models whichare Gibbs perturbations of the Haar measure, and express the expansioncoefficients recursively in terms of the unique solution of a noncommutativeinitial value problem. The recursion obtained is closely related to the"topological recursion" which underlies the asymptotics of many random matrixensembles and appears in diverse enumerative geometry problems, but has notpreviously appeared in the context of random unitary matrices. Our approachconsists of two main ingredients: an asymptotic study of the Schwinger-Dysonlattice over noncommutative Laurent polynomials, and uniform control on thecumulants of Gibbs measures on product unitary groups. The required cumulantbounds are obtained by concentration of measure arguments.

Asymptotics of unitary multimatrix models: The Schwinger-Dyson lattice and topological recursion