Casimir Scaling of domain wall tensions in the deconfined phase of D=3+1 SU(N) gauge theories

We perform lattice calculations of the spatial 't Hooft k-string tensions in the deconfined phase of SU(N) gauge theories for N=2,3,4,6. These equal (up to a factor of T) the surface tensions of the domain walls between the corresponding (Euclidean) deconfined phases. For T much larger than T_c our results match on to the known perturbative result, which exhibits Casimir Scaling, being proportional to k(N-k). At lower T the coupling becomes stronger and, not surprisingly, our calculations show large deviations from the perturbative T-dependence. Despite this we find that the behaviour proportional to k(N-k) persists very accurately down to temperatures very close to T_c. Thus the Casimir Scaling of the 't Hooft tension appears to be a `universal' feature that is more general than its origin in the low order high-T perturbative calculation. We observe the `wetting' of these k-walls at T around T_c and the (almost inevitable) `perfect wetting' of the k=N/2 domain wall. Our calculations show that as T tends to T_c the magnitude of the spatial `t Hooft string tension decreases rapidly.