Local Limit of Random Regular Bipartite Planar Maps

We prove the existence of the local limit of uniform random d-regular bipartite planar maps, for every $d\ge 3$, 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.