It is proved that the potential kernel of a recurrent, aperiodic
random walk on the integer lattice $\mathbb{Z}^2$ admits an asymptotic
expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const +
|x|^{-1} U_1 (\omega^x) + |x|^{-2} U_2 (\omega^x) + \dots ,$$ where $|Q|$ and
$Q(\theta)$ are, respectively, the determinant and the quadratic form of the
covariance matrix of the increment X of the random walk, $\omega^x =
x/|x|$ and the $U_k (\omega)$ are smooth functions of $\omega, |\omega| = 1$,
provided k that all the moments of X are finite. Explicit forms of $U_1$
and $U_2$ are given in terms of the moments of X.