  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. 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 $q$.
