Mathematics – Probability
Scientific paper
2011-08-10
Mathematics
Probability
36 pages
Scientific paper
We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on Z^2. In more interesting versions, a p fraction of customers orders the "freshest available" product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on p. We then turn our attention to the critical Fortuin-Kastelyn random planar map model, which gives, for each q>0, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the q-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on q. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at p = 1/2, q=4.
No associations
LandOfFree
Quantum gravity and inventory accumulation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Quantum gravity and inventory accumulation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quantum gravity and inventory accumulation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-665920