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In this paper we consider the natural random walk on a planar graph and scaleit by a small positive number $\delta$. Given a simply connected domain $D$ andits two boundary points $a$ and $b$, we start the scaled walk at a vertex ofthe graph nearby $a$ and condition it on its exiting $D$ through a vertexnearby $b$, 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 walkand the dual walk of it.

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve