Recurrence of the Uniform Infinite Half-Plane Map via duality of resistances

TL;DR

This paper proves the recurrence of simple random walk on the Uniform Infinite Half-Plane Map and establishes a logarithmic lower bound on resistance growth, using duality and peeling techniques.

Contribution

It introduces a novel resistance bound for the Uniform Infinite Half-Plane Map, improving upon previous recurrence proofs for planar maps.

Findings

Random walk on the map is recurrent.

Resistance between root and boundary grows at least logarithmically.

Resistance bound is expected to be sharp.

Abstract

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.