  The site percolation problem is studied on d-dimensional generalisations of
the Kagome' lattice. These lattices are isotropic and have the same
coordination number q as the hyper-cubic lattices in d dimensions, namely q=2d.
The site percolation thresholds are calculated numerically for d= 3, 4, 5, and
6. The scaling of these thresholds as a function of dimension d, or
alternatively q, is different than for hypercubic lattices: p_c ~ 2/q instead
of p_c ~ 1/(q-1). The latter is the Bethe approximation, which is usually
assumed to hold for all lattices in high dimensions. A series expansion is
calculated, in order to understand the different behaviour of the Kagome'
lattice. The return probability of a random walker on these lattices is also
shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices
these results imply p_c ~ 1/(q-1).
