Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit

The paper deals with the discrete Laplacian on a uniform infinite square lattice. The definition of its fundamental solution or lattice Green function (LGF) is clarified as the Fourier coefficients of a certain generalized periodic function g. Such a functional must be regularized and gives the LGF up to a constant equal to < g >, the mean value of g. For < g >=0, the LGF may be expressed in an exact analytic form in terms of hypergeometric and gamma functions. The continuum limit of the LGF is finally studied requiring an appropriate renormalization of < g > in order to obtain the logarithmic Coulomb potential.