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In perturbative expansion of field theories on a non-commutative geometry, itis known that planar diagrams dominate when the non-commutativity parameter$\theta$ goes to infinity. We discuss whether the ``planar dominance'' occursalso in the case where $\theta$ is finite, but the external momentum goes toinfinity instead. While this holds trivially at the one-loop level, it is notobvious at the two-loop level in particular in the presence of UV divergences.We perform explicit two-loop calculations in the six-dimensional $\phi^3$theory, and confirm that nonplanar diagrams after renormalization do vanish inthe above limit.Comment: 14 pages, 7 Figure

Planar Dominance in Non-Commutative Field Theories at Infinite External Momentum