September 2022 Recurrence of the uniform infinite half-plane map via duality of resistances
Thomas Budzinski, Thomas Lehéricy
Author Affiliations +
Ann. Probab. 50(5): 1725-1754 (September 2022). DOI: 10.1214/21-AOP1539


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 logr. 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.

Funding Statement

Both authors acknowledge the support of ERC Advanced Grant 740943 GeoBrown.


The first author thanks Nicolas Curien and Asaf Nachmias for useful discussions.


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Thomas Budzinski. Thomas Lehéricy. "Recurrence of the uniform infinite half-plane map via duality of resistances." Ann. Probab. 50 (5) 1725 - 1754, September 2022.


Received: 1 February 2020; Revised: 1 May 2021; Published: September 2022
First available in Project Euclid: 24 August 2022

MathSciNet: MR4474500
zbMATH: 1500.05056
Digital Object Identifier: 10.1214/21-AOP1539

Primary: 05C80
Secondary: 60D05

Keywords: half-plane map , Planar map , resistance , self-duality , Tutte bijection

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 5 • September 2022
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