# Basic properties of the infinite critical-FK random map

### Linxiao Chen

Université Paris-Sud, Orsay, France and CEA Saclay, Gif-sur-Yvette, France

## Abstract

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 $q$.

## Cite this article

Linxiao Chen, Basic properties of the infinite critical-FK random map. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 4 (2017), no. 3, pp. 245–271

DOI 10.4171/AIHPD/40