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