Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications

The lattice Green functions for the discrete planar Laplacian defined on regular square lattice wrapped around cylinders and tori are rigorously defined and obtained in an exact analytic form. The method of images well-known in potential theory is implemented to derive for many other geometries with free boundaries (semi-infinite or finite cylinders and strips, rectangle) the related exact lattice Green-Neumann functions needed to readily solve discrete Neumann problems or, via a Neumann-to-Dirichlet mapping, discrete Dirichlet problems for these flat square lattices. Some applications are thus proposed as explicit expressions of two-point resistances for related resistor networks, and some probability-based characteristics regarding the associated Pòlya’s random walks.

• Publication year

  2023

• Venue

  Journal of Physics A: Mathematical and Theoretical

• Publication date

  2023-04-24

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1088/1751-8121/accfd5
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Impedance responses and size-dependent resonances in topolectrical circuits via the method of images

  2022cited by this paper
• Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit

  2021cited by this paper
• Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya

  2020cited by this paper
• Unconventional superconductivity discovered in graphene bilayers

  2018cited by this paper
• Simple Models of Lateral Heterostructures

  2018cited by this paper
• Statistical Field Theory

  2014cited by this paper
• Fundamental solution of the Laplacian on flat tori and boundary value problems for the planar Poisson equation in rectangles

  2014cited by this paper
• Graphene: Carbon in Two Dimensions

  2012cited by this paper
• Lattice gauge theories

  2012cited by this paper
• Lattice Green's functions in all dimensions

  2010cited by this paper
• Asymptotic expansion for the resistance between two maximally separated nodes on an M by N resistor network.

  2010cited by this paper
• INFINITE NETWORKS OF IDENTICAL CAPACITORS

  2009cited by this paper
• The exact evaluation of the corner-to-corner resistance of an M × N resistor network: asymptotic expansion

  2008cited by this paper
• Graphene: Exploring carbon flatland

  2007cited by this paper
• Diffusion Limited Reactions

  2007cited by this paper
• Generalized Functions

  2006cited by this paper
• On the resistance of an infinite square network of identical resistors – Theoretical and experimental comparison

  2006cited by this paper
• Theory of impedance networks: the two-point impedance and LC resonances

  2006cited by this paper
• Discrete scattering theory : Green's function for a square lattice

  2006cited by this paper
• Exact expressions of mean first-passage times and splitting probabilities for random walks in bounded rectangular domains.

  2006cited by this paper
• Capacitance between two points on an infinite grid

  2005cited by this paper
• First-passage times for random walks in bounded domains.

  2005influential reference
• Solution of Dirichlet Problems with Discrete Double-Layer Potentials

  2005cited by this paper
• Random walks and effective resistances on toroidal and cylindrical grids

  2004cited by this paper
• Theory of resistor networks : The two-point resistance

  2004cited by this paper
• Series expansions for lattice Green functions

  2000cited by this paper
• Infinite resistive lattices

  1999cited by this paper
• Application of the lattice Green's function for calculating the resistance of an infinite network of resistors

  1999cited by this paper
• Lattice discretization in quantum scattering

  1996cited by this paper
• The Dirichlet to Neumann map for a resistor network

  1991cited by this paper
• Elements of Green's Functions and Propagation: Potentials, Diffusion, and Waves

  1989cited by this paper
• The electrical resistance of a graph captures its commute and cover times

  1989cited by this paper
• Quarks, Gluons and Lattices: Cambridge Monographs on Mathematical Physics

  1986influential reference
• Quarks, Gluons and Lattices

  1984cited by this paper
• A Table of Series and Products

  1977cited by this paper
• Equations of mathematical physics

  1972cited by this paper
• 強磁性体中の不純物スピン (Lattice Green's Function)

  1971cited by this paper
• Lattice Green's Function. Introduction

  1971cited by this paper
• Random walk and electric currents in networks

  1959cited by this paper
• Discrete potential theory

  1953cited by this paper
• Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz

  1921cited by this paper

• Topolectrical circuits—Recent experimental advances and developments

  2025cites this paper
• Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

  2024influential citation