Using a geometruc 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 (N CS 2) grows with the spatial volume. We also find numerical evidence that tunneling in 4d is increased at high temperature.
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