In this paper we investigate the critical Fortuin–Kasteleyn (cFK) random map model. For each and integer , this model chooses a planar map of edges with a probability proportional to the partition function of critical -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 , 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 , and mutually singular in distribution for different values of .
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. 245–271DOI 10.4171/AIHPD/40