Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors

We calculate the resistance between two arbitrary grid points of several infinite lattice structures of resistors by using the lattice Green's function. The resistance for $d$ dimensional hypercubic, rectangular, triangular and honeycomb lattice of resistors are discussed in details. We give a recurrence formulas for the resistance between arbitrary lattice points of the square lattice. For large separation between nodes we obtain the asymptotic form of the resistance for square lattice and the finite and exact value of the resistance for simple cubic lattice. We point out the relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian. Our Green's function method can be applied in a straightforward manner for other type of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.