Loop-erased walks and total positivity

We consider matrices whose elements enumerate weights of walks in planar di rected weighted graphs (not necessarily acyclic). These matrices are totall y nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total posi tivity of hitting matrices of Brownian motion in planar domains.