Application of the lattice Green's function for calculating the resistance of an infinite grid 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 defined on the lattice. The resistance for $d$ dimensional hypercubic, rectangular, triangular and honeycomb lattice of resistors are discussed in details. For large separation between the points where the current enters and exits we obtain the asymptotic form of the resistance in case of 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 defined on the lattice. Our general 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 solid state physics.