Size effects in the electronic properties of finite arrays of exchange   coupled quantum dots: A renormalization group approach
  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 multi-stage 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. 
Jiaxiang Wang, Sabre Kais, F. Remacle and R. D. Levine