  Interpreting the way that the SU(3) bare lattice coupling runs with the
lattice spacing is complicated by the fact that there is a smooth cross-over
region in which the strong coupling expansion transforms into a weak-coupling
one. For N > 4, however, there is a first order bulk transition that cleanly
separates the strong and weak coupling regimes. We find that in this case the
calculated string tension can be readily fitted throughout the weak coupling
region by a standard 3-loop expression modified by lattice spacing corrections
of the expected form. While our fits demand the presence of the latter, they do
not constrain the perturbative coupling scheme enough to enable us to extract a
usefully accurate value of a(beta) in units of Lambda_MSbar. To resolve this
ambiguity we turn to SU(3) where we use the Schrodinger Functional coupling
scheme to extract a value of r_0 x Lambda_SF as a benchmark. We then find that
the Parisi mean-field improved coupling scheme closely reproduces this result.
We also develop a comparison between different schemes that does not rely on
the calculation of any physical quantity and which can therefore be applied
much further into weak coupling. Again the Parisi scheme is favoured over the
others that we compare. Using the mean-field scheme we have fitted the values
of the string tension that have been calculated for N=2 to N=8, to obtain
Lambda_MSbar/sqrt(sigma) = 0.503(2)(40) + 0.33(3)(3)/N^2 for N > 2, where the
first error is statistical and the second is our estimate of the systematic
error from all sources.
