In this paper we consider the 'natural' random walk on a planar graph and scale it by a small positive number δ. 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 δ ↓ 0, to the chordal SLE2 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. Our result is an extension of one due to Dapeng Zhan  where the problem is considered on the square lattice. A convergence to the radial SLE2 has been obtained by Lawler, Schramm and Werner  for the square and triangular lattices and by Yadin and Yehudayoff  for a wide class of planar graphs. Our proof, though an adaptation of that of  and , involves some new ingredients that arise from two sources: one for dealing with a martingale observable that is different from that used in  and  and the other for estimating the harmonic measures of the random walk started at a boundary point of a domain.
"Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve." Kodai Math. J. 37 (2) 303 - 329, June 2014. https://doi.org/10.2996/kmj/1404393889