  We find a strong-to-weak coupling cross-over in D=2+1 SU(N) lattice gauge
theories that appears to become a third-order phase transition at N=\infty, in
a similar way to the Gross-Witten transition in the D=1+1 SU(N\to\infty)
lattice gauge theory. There is, in addition, a peak in the specific heat at
approximately the same coupling that increases with N, which is connected to
Z_N monopoles (instantons), reminiscent of the first order bulk transition that
occurs in D=3+1 for N > 4. Our calculations are not precise enough to determine
whether this peak is due to a second-order phase transition at N=\infty or to a
third-order phase transition with different critical behaviour to that of the
Gross-Witten transition. We investigate whether the trace of the Wilson loop
has a non-analyticity in the coupling at some critical area, but find no
evidence for this. However we do find that, just as one can prove occurs in
D=1+1, the eigenvalue density of a Wilson loop forms a gap at N=\infty at a
critical value of its trace. We show that this gap formation is in fact a
corollary of a remarkable similarity between the eigenvalue spectra of Wilson
loops in D=1+1 and D=2+1 (and indeed D=3+1): for the same value of the trace,
the eigenvalue spectra are nearly identical. This holds for finite as well as
infinite N; irrespective of the Wilson loop size in lattice units; and for
Polyakov as well as Wilson loops.
