  Matrix field theory is a combinatorially non-local field theory which has
recently been found to be a non-trivial but solvable QFT example. To generalize
such non-perturbative structures to other models, a more combinatorial
understanding of Dyson-Schwinger equations and their solutions is of high
interest. To this end we consider combinatorial Dyson-Schwinger equations
manifestly relying on the Hopf-algebraic structure of perturbative
renormalization. We find that these equations are fully compatible with
renormalization, relying only on the superficially divergent diagrams which are
planar ribbon graphs, i.e. decompleted dual combinatorial maps. Still, they are
of a similar kind as in realistic models of local QFT, featuring in particular
an infinite number of primitive diagrams as well as graph-dependent
combinatorial factors.
