Basic properties of the infinite critical-FK random map
Abstract
We investigate the critical Fortuin-Kasteleyn (cFK) random map model. For each q∈ [0,∞] and integer n≥ 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. Sheffield introduced the hamburger-cheeseburger 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.