Exact Results for a Body-Centered Cubic Lattice Green's Function with Applications in Lattice Statistics. I

Abstract

In this paper the body‐centered cubic lattice Green's function $$\mathrm{P(}\mathbf{I}\mathrm{,z)=}\frac{1}{{\pi }^3}\int {\displaystyle {\int }_0^{\pi }}\text{∫(1−z}\cos x_1\cos x_2\cos x_3{)}^{\text{−1}}\cos l_1{x}_1\cos l_2{x}_2\cos l_3{x}_3{\mathrm{dx}}_1{\mathrm{dx}}_2{\mathrm{dx}}_3$$ , where l₁, l₂, and l₃ are all even, or all odd, is studied. A complete analytic continuation for P(z) ≡ P(0, z) is derived of the form $$\mathrm{P(z)=}{\displaystyle \sum _{\text{n=0}}^{\infty }}{B}_n{\text{(1−z}}^2{)}^n{\text{−(1−z}}^2{)}^{\frac{1}{2}}{\displaystyle \sum _{\text{n=0}}^{\infty }}{C}_n{\text{(1−z}}^2{)}^n$$ , where |1 − z²| < 1. Explicit formulas, recurrence relations, and asymptotic expansions are established for the coefficients B_n and C_n. A similar analytic continuation in powers of 1 − z is also investigated. The generalized Watson integral $$\mathrm{I(m,n)=}\frac{1}{{\pi }^3}\int {\displaystyle {\int }_0^{\pi }}\text{∫(1−}\cos x_1\cos x_2\cos x_3{)}^{\text{−1}}{\cos }^{\text{2m}}{x}_1{\cos }^{\text{2n}}{x}_2{\mathrm{dx}}_1{\mathrm{dx}}_2{\mathrm{dx}}_3$$ , where m ≥ 0 and n ≥ 0, is evaluated in closed form. Using this result, we show that P(I, 1) can, in principle, be evaluated for arbitrary I. Exact expressions and numerical values for P(I, 1) are given for 0 ≤ l₁ ≤ l₂ ≤ l₃ ≤ 8. Detailed applications of the above results are made in the theory of random walks on a body‐centered cubic lattice. In particular, a new asymptotic expansion for the expected number of distinct lattice sites visited during an n‐step random walk is obtained. The closely related Green's function $$\lim {\displaystyle {lim}_{\text{ε→0+}}}\frac{1}{{\pi }^3}\int {\displaystyle {\int }_0^{\pi }}{\mathrm{\int (\xi }}_0\mathrm{-i\epsilon -}\cos x_1\cos x_2\cos x_3{)}^{\text{−1}}{\mathrm{dx}}_1{\mathrm{dx}}_2{\mathrm{dx}}_3$$ , where ξ₀ is real, is expressed in terms of complete elliptic integrals for all ξ₀ > 0, and evaluated numerically in the range 0 < ξ₀ ≤ 1. The behavior of this Green's function in the neighborhood of the singularities at ξ₀ = 0 and 1 is also discussed. No attempt is made, in the present paper, to discuss P(I, z) for the general case I ≠ 0 and z ≠ 1.