Recurrence of the uniform infinite half-plane map via duality of resistances

We study the simple random walk on Uniform Infinite Half-Plane Map, which is local limit of critical Boltzmann planar maps with a large and boundary. prove that recurrent, resistance between root boundary hull radius r at least order logr. This bound expected to be sharp, better than those following from previous proofs recurrence for nonbounded-degree models. Our main tools are self-duality uniform maps, classical lemma about duality resistances some peeling estimates. The proof shares ideas Russo–Seymour–Welsh theory in percolation.