Quantum gravity and inventory accumulation
Name
1108.2241.pdf
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1.08 MB
Format
Adobe PDF
Checksum (MD5)
8f7437bfae094d227a22d15fe2b71f58
Author(s)
Sheffield, Scott Roger
Date Issued
November 2016
Journal
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Citation
Sheffield, Scott. “Quantum Gravity and Inventory Accumulation.” The Annals of Probability 44, 6 (November 2016): 3804–3848
Version
Author's final manuscript
Abstract
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 Z². 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1214/15-AOP1061
