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

Abstract

We calculate the resistance between two arbitrary grid points of several infinite lattice structures of resistors by using lattice Green's functions. The resistance for $d$ dimensional hypercubic, rectangular, triangular and honeycomb lattices of resistors is discussed in detail. We give recurrence formulas for the resistance between arbitrary lattice points of the square lattice. For large separation between nodes we calculate the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a 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 to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.

Cited by in corpus (59)

• Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback
• Theory of resistor networks: The two-point resistance
• Simulating hyperbolic space on a circuit board
• Theory of impedance networks: The two-point impedance and LC resonances
• Lattice Green functions in all dimensions
• Perturbation of infinite networks of resistors
• Recursion-transform method on computing the complex resistor network with three arbitrary boundaries
• The two-point resistance of a cobweb with a superconducting boundary
• Uniform tiling with electrical resistors
• Exact Evaluation of The Resistance in an Infinite Face- Centered Cubic Network
• The exact evaluation of the corner-to-corner resistance of an M x N resistor network: Asymptotic expansion
• Scaling in Small-World Resistor Networks
• A generalised formulation of the Laplacian approach to resistor networks
• Error Scaling Laws for Linear Optimal Estimation from Relative Measurements
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm
• Quasiparticle interference from magnetic impurities
• A parallel fast multipole method for elliptic difference equations
• Abelian Sandpile Model on the Honeycomb Lattice
• Impedance responses and size-dependent resonances in topolectrical circuits via the method of images
• Hitting and commute times in large graphs are often misleading
• Anomalous fractal scaling in two-dimensional electric networks
• Evaluation of effective resistances in pseudo-distance-regular resistor networks
• The effective resistance of the $N$-cycle graph with four nearest neighbors
• On equivalent resistance of electrical circuits
• Electronic transport on carbon nanotube networks: a multiscale computational approach
• Resiliently evolving supply-demand networks
• Green's functions and DOS for some 2D lattices
• Long-range correlations in a locally driven exclusion process
• Pattern formation in fast-growing sandpiles
• Infinite Body Centered Cubic Network of Identical Resistors
• Geometric dependence of Nb-Bi${}_2$Te${}_3$-Nb topological Josephson junction transport parameters
• The shape of the reliability polynomial of a hammock network
• Calculating effective resistances on underlying networks of association schemes
• A chaotic lattice field theory in one dimension
• Green's function of a finite chain and the discrete Fourier transform
• Infinite Networks of Identical Capacitors
• Three-leg correlations in the two component spanning tree on the upper half-plane
• Planar potential flow on Cartesian grids
• Keplerian Dynamics on the Heisenberg Group and Elsewhere
• Evaluating transport in irregular pore networks
• Current Redistribution in Resistor Networks: Fat-Tail Statistics in Regular and Small-World Networks
• Crossover exponents, fractal dimensions and logarithms in Landau-Potts field theories
• Effective resistances of two dimensional resistor networks
• Sandpile probabilities on triangular and hexagonal lattices
• Proportionate growth in patterns formed in the rotor-router model
• Green's function on lattices
• A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory
• A general method for calculating lattice Green functions on the branch cut
• Lattice Green's Functions for High Order Finite Difference Stencils
• Electrical conductivity of conductive films based on random metallic nanowire networks
• General analytical solutions for DC/AC circuit network analysis
• Differential Equation Approach for One- and Two- Dimensional Lattice Green's Function
• Average resistance of toroidal graphs
• General theory of perturbation of infinite resistor networks
• Line defects in infinite networks of resistors
• Remarks on the lattice Green's Function for the anisotropic Face Centered Cubic Lattice
• Infinite Network of Identical Capacitors by Green's Function
• Finite and infinite product transformations