  Using a geometric definition for the lattice Chern-Simons term in even
dimensions, we have studied the distribution of Chern-Simons numbers for the
2d-U(1) and the 4d-SU(2) lattice Higgs models. The periodic structure of the
distributions is preserved in our lattice formulation and has been examined in
detail. In both cases the finite size effects visible in the distribution of
Chern-Simons numbers are well accounted for by the Haar measure. Moreover, we
find that $\langle N_{CS}^2 \rangle$ grows with the spatial volume. We also
find numerical evidence that tunneling in 4d is increased at high temperature.
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