随机平面映射II上临界Fortuin-Kastelyn模型的标度极限：局部估计和空约化字指数
Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent
摘要：
我们继续研究Sheffield（2011）引入的库存累积，该模型编码了一个随机平面映射，该映射由从临界Fortuin-Kasteleyn（FK）模型中采样的循环集合装饰。我们证明了库存累积模型的各种\emph{local estimates}，即，从模型中采样的简化单词中给定类型的符号的精确数量的估计。使用我们的估计，我们得到了相关的二维随机游动的标度极限的条件下，它停留在第一象限的一个单位时间，并最终在一个特定的位置在内部的第一象限。我们还获得了从库存累积模型中采样的长度为$2n$的字对应于空的约简字的概率的指数，这等价于临界FK平面映射模型的分拆函数的渐近公式。本文的估计将在随后的论文中使用，以获得与有限体积FK平面映射相关联的晶格行走的缩放极限。
英文摘要：
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.
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