Potential kernel for two-dimensional random walk

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice Z(2) admits an asymptotic expansion of the form (2 pi root\Q\)(-1)ln Q(x(2), - x(1)) + const + \x\U--1(1)(omega(x)) + \x\U--2(2)(omega(2)) + ..., 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(2) = x/\x\ and the U-h(omega) are smooth functions of omega, \omega\ = 1, provided 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.