Lattice Green's Function. Introduction

Abstract

Physical, analytical, and numerical properties of the lattice Green's functions for the various lattices are described. Various methods of evaluating the Green's functions, which will be developed in the subsequent papers, are mentioned.

Topologically correct phase boundaries and transition temperatures for Ising Hamiltonians via self-consistent coarse-grained cluster-lattice models

“…Although the integrand contains a singularity at k=0, the integral (a lattice Green's function 40 ) is convergent 41 for cubic lattices and the numerical values are given in [37]. We have obtained them via numerical integration using MATLAB.…”

Coulomb interactions via local dynamics: a molecular-dynamics algorithm

“…A lot is known about this function, in particular in the limit L i → ∞ [15,16,17]. For our purposes, however, it is sufficient to note that (i) G can be calculated at the beginning of the simulation once and for all, including the finite size effect, and that (ii) G( r = 0) is finite, even in the limit L i → ∞ (but, of course, keeping a fixed).…”

Relations Between Resistance Distances of a Graph and its Complement or its Contraction

“…5 Besides, it has been shown that effective resistance is also relevant to a wide range of problems ranging from random walks on graphs, 6 the theory of harmonic functions 7 to lattice Green's functions. 8 However, not until the contribution of Klein and Randić, 1 it is revealed that the effective resistance is an intrinsic graph metric. Thanks to this important fact, people came to realize that resistance distance is a fundamental concept of graph theory and began to consider its mathematical properties as well as its applications in chemistry.…”