  The Mott insulator-to-superfluid transition exhibited by the Bose-Hubbard
model on a two-dimensional square lattice occurs for any value of the chemical
potential, but becomes critical at the tips of the so-called Mott lobes only.
Employing a numerical approach based on a combination of high-order
perturbation theory and hypergeometric analytic continuation we investigate how
quantum critical properties manifest themselves in computational practice. We
consider two-dimensional triangular lattices and three-dimensional cubic
lattices for comparison, providing accurate parametrizations of the phase
boundaries at the tips of the respective first lobes. In particular, we lend
strong support to a recently suggested inequality which bounds the divergence
exponent of the one-particle correlation function in terms of that of the
two-particle correlation function, and which sharpens to an equality if and
only if a system becomes critical.
