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