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# Annales de l’Institut Henri Poincaré D

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**Volume 4, Issue 3, 2017, pp. 245–271**

**DOI: 10.4171/AIHPD/40**

Published online: 2017-09-26

Basic properties of the infinite critical-FK random map

Linxiao Chen^{[1]}(1) Université Paris-Sud, Orsay, France and CEA Saclay, Gif-sur-Yvette, France

In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each $q \in [0, \infty]$ and integer $n \geq 1$, this model chooses a planar map of $n$ edges with a probability proportional to the partition function of critical $q$-Potts model on that map. Sheeld introduced the hamburger–cheeseburer bijection which maps the cFK random maps to a family of random words, and remarked that one can construct infinite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When $q = 1$, this provides an alternative construction of the UIPQ. In addition, we show that the limit is almost surely one-ended and recurrent for the simple random walk for any $q$, and mutually singular in distribution for different values of $q$.

*Keywords: *Fortuin–Kasteleyn percolation, random planar maps, hamburger–cheeseburer bijection, local limits, recurrent graph, ergodicity of random graphs

Chen Linxiao: Basic properties of the infinite critical-FK random map. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 4 (2017), 245-271. doi: 10.4171/AIHPD/40