Local Limit of Random Regular Bipartite Planar Maps

TL;DR

This paper establishes the local limit for uniform random d-regular bipartite planar maps as the number of vertices grows, using a bijection with blossoming trees and extending to infinite trees.

Contribution

It proves the existence of the local limit for these maps for all d ≥ 3, extending previous bijections to infinite trees and analyzing their properties.

Findings

The local limit exists for all d ≥ 3.

The limit object is almost surely one-ended.

The limiting map is recurrent for simple random walk.

Abstract

We prove the existence of the local limit of uniform random d-regular bipartite planar maps, for every $d\geq 3$, as the number of vertices tends to infinity. The proof relies on a bijection between maps and so-called blossoming trees established in a previous work. After proving local convergence of the associated decorated trees, we extend the bijection to infinite trees and transfer the convergence to planar maps. The limiting object is almost surely one-ended and recurrent for the simple random walk.