Transport properties of arrays of metallic quantum dots are governed by the distance-dependent exchange coupling between the dots. It is shown that the effective value of the exchange coupling, as measured by the charging energy per dot, depends monotonically on the size of the array. The effect saturates for-hexagonal arrays of over 7(5) unit cells. The discussion uses a multistage block renormalization group approach applied to the Hubbard Hamiltonian. A first-order phase transition occurs upon compression of the lattice, and the size dependence is qualitatively different for the two phases.