  We study the Doob's $h$-transform of the two-dimensional simple random walk
with respect to its potential kernel, which can be thought of as the
two-dimensional simple random walk conditioned on never hitting the origin. We
derive an explicit formula for the Green's function of this random walk, and
also prove a quantitative result on the speed of convergence of the
(conditional) entrance measure to the harmonic measure (for the conditioned
walk) on a finite set.
