Exact results for the anisotropic honeycomb lattice Green function with applications to three-step Pearson random walks

Abstract

The mathematical properties of the lattice Green function for the anisotropic honeycomb lattice are studied, where is a complex variable which lies in a plane, and is a real anisotropy parameter with . This double integral defines a single-valued analytic function provided that a cut is made along the real axis from to . In order to analyse the behaviour of along the edges of the cut it is convenient to define the limit function where . It is proved that the limit functions and can be sectionally evaluated exactly for all , in terms of various elliptic integrals of the first kind , where is a rational function of and u.Next, it is demonstrated that is a solution of a second order linear differential equation with eight ordinary regular singular points and two apparent singular points. It is shown that the apparent singularities can be removed by constructing a particular differential equation of third order. The series solution where and is investigated. In particular, we show that, in general, satisfies a four-term linear recurrence relation. This result is used to determine the asymptotic behaviour of as .Integral representations are established for and . It is found that where J0(z) and Y0(z) denote Bessel functions of the first and second kind, respectively, and .Finally, the results are applied to the lattice Green function for the anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane.