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

We prove the existence of a 1/N expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the "topological recursion" which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems, but has not previously appeared in the context of random unitary matrices. Our approach consists of two main ingredients: an asymptotic study of the Schwinger-Dyson lattice over noncommutative Laurent polynomials, and uniform control on the cumulants of Gibbs measures on product unitary groups. The required cumulant bounds are obtained by concentration of measure arguments.