  In this paper we consider the natural random walk on a planar graph and scale
it by a small positive number $\delta$. Given a simply connected domain $D$ and
its two boundary points $a$ and $b$, we start the scaled walk at a vertex of
the graph nearby $a$ and condition it on its exiting $D$ through a vertex
nearby $b$, and prove that the loop erasure of the conditioned walk converges,
as $\delta \downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in
$D$, provided that an invariance principle is valid for both the random walk
and the dual walk of it.
