Are the Tonks regimes in the continuum and on the lattice truly equivalent?

Motivated by recent experiments, we compare the Tonks (i.e. hard-core boson gas) regime achieved in an optical lattice with the Tonks regime of a one-dimensional Bose gas in the continuum. For the lattice gas, we compute the local (i.e. on-site) two-body correlations as a function of temperature and the filling of the lattice. It is found that this function saturates to a constant value with increasing temperature. Furthermore, the parameter that characterizes the long-distance correlations in the lattice Tonks regime is also obtained, showing that on the lattice the long-distance correlations enter the Tonks regime more rapidly than in the continuum.