Potential kernel for two-dimensional random walk

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice Z² admits an asymptotic expansion of the form (2 π √|Q|)⁻¹ ln Q(x₂, -x₁) + + |x|⁻¹ U₁ (ωˣ) + |x|⁻² U₂ (ωˣ) + , where $|Q|$ and Q(θ) are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, ωˣ = x/|x| and the Uₖ (ω) are smooth functions of ω, |ω| = 1, provided k that all the moments of X are finite. Explicit forms of U₁ and U₂ are given in terms of the moments of X.