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

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent