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This paper concerns a random walk on a planar graph and presents certainestimates concerning the harmonic measures for the walk in a grid domain whichestimates are useful for showing the convergence of a LERW (loop-erased randomwalk) to an SLE (stochastic Loewner evolution). We assume that the walk startedat a fixed vertex of the graph satisfies the invariance principle as in Yadinand Yehudayoff [16] in which the convergence of LERW to a radial SLE isestablished in this setting. Our main concern is chordal case, where a randomwalk is started at a boundary vertex of a simply connected grid domain andconditioned to exit it through another boundary vertex specified in advance.The primary contribution of the present paper is an estimate, which states thatthe excursion of the conditioned walk leaves an intrinsic neighborhood of itsinitial point not 'along' the boundary but through an intrinsic interior of thedomain with high probability. Based on this result we give a proof for theconvergence to the chordal SLE, a result that has recently been proved bySuzuki [12] under an analyticity assumption on the boundary of the domainarising in the limit.

Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$