Inventory Accumulation with $k$ Products

Abstract

Sheffield (2011) proposed an inventory accumulation model with two types of products to study the critical Fortuin-Kasteleyn model on a random planar map, and showed that a two-dimensional inventory accumulation trajectory in the discrete model scales to a correlated planar Brownian motion. In this work, we generalize the inventory model to $k$ types of products for any integer $k\ge 2$, and prove that the corresponding trajectory scales to a $k$-dimensional Brownian motion with an appropriate covariance matrix.

arXiv comment16 pages

References in corpus (4)

• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case
• Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent
• Sandpiles and unicycles on random planar maps