  We continue our study of the inventory accumulation introduced by Sheffield
(2011), which encodes a random planar map decorated by a collection of loops
sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various
\emph{local estimates} for the inventory accumulation model, i.e., estimates
for the precise number of symbols of a given type in a reduced word sampled
from the model. Using our estimates, we obtain the scaling limit of the
associated two-dimensional random walk conditioned on the event that it stays
in the first quadrant for one unit of time and ends up at a particular position
in the interior of the first quadrant. We also obtain the exponent for the
probability that a word of length $2n$ sampled from the inventory accumulation
model corresponds to an empty reduced word, which is equivalent to an
asymptotic formula for the partition function of the critical FK planar map
model. The estimates of this paper will be used in a subsequent paper to obtain
the scaling limit of the lattice walk associated with a finite-volume FK planar
map.
