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Recurrence of the uniform infinite half-plane map via duality of resistances. (English) Zbl 1500.05056

Summary: 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 nonbounded-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.

MSC:

05C81 Random walks on graphs
05C80 Random graphs (graph-theoretic aspects)
05C10 Planar graphs; geometric and topological aspects of graph theory
60D05 Geometric probability and stochastic geometry
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