## Quantum gravity and inventory accumulation

##### Author(s)

Sheffield, Scott Roger
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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.

##### Date issued

2016-11##### Department

Massachusetts Institute of Technology. Department of Mathematics##### 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

##### ISSN

0091-1798