Basic properties of the infinite critical-FK random map

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 .