2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59018We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally 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 positivity of hitting matrices of Brownian motion in planar domains.20 pages; 7 figuresCombinatoricsProbabilityRings and Algebras15A48 (Primary) 05C50, 31C20, 60J65 (Secondary}text