Is Perturbation Theory the Asymptotic Expansion in Lattice Gauge Theories?

It is shown that in a gauge theory on an $L^d$ lattice with a compact Lie group, the weak-coupling expansion of any gauge-invariant Green's function may cease to be an asymptotic representation of the true answer as $L\to \infty$. The disagreement is expected to occur at the two-loop level, in non-Abelian models.