Exact results for the diamond lattice Green function with applications to uniform random walks in a plane

Abstract

The mathematical properties of the diamond lattice Green functionGD(wd)=wdπ3∫0π∫0π∫0πdθ1dθ2dθ3wd2−4[1+c(θ1)c(θ2)+c(θ2)c(θ3)+c(θ3)c(θ1)]are investigated, where and is a complex variable which lies in a plane. This triple 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 functionlimϵ→0+GD(ud±iϵ)≡GRD(ud)∓iGID(ud),where .It is shown that and can be evaluated exactly for all in terms of various hypergeometric functions, where the argument function is always real-valued and rational. The third-order linear Fuchsian differential equation satisfied by is also used to derive series expansions for and which are valid in the neighbourhood of the regular singular points and 4. Integral representations are established for and , where . In particular, it is proved that2πGRD(x)=∫0∞xtY0(xt)[J0(t)]4dt,2πGID(x)=∫0∞xtJ0(xt)[J0(t)]4dt,where and denote Bessel functions of the first and second kind, respectively.Finally, the new results for are applied to the problem of Pearson random walks in a plane when each walk consists of four steps of equal length. It is found that the radial probability density function for this case is given byp4(x)=(x2)13{p4(2)2F1[16,16;56;ξ(x)]2+3(x2−4)2π4p4(2)(2x2)132F1[13,13;76;ξ(x)]2}−H(x2−4)2π2x2−42F1[16,16;56;ξ(x)]2F1[13,13;76;ξ(x)],where , , is the Heaviside step function,p4(2)=6π3K[122(3−1)]2and denotes a complete elliptic integral of the first kind.