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.Comment: 14 pages, 6 figures. v2: Fixed the proof of main theorem, removed old
lemma 5, added results on mutually singular measures and ergodicity.
Submitted to Annales de l'Institut Henri Poincar\'e
