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November 2016 Quantum gravity and inventory accumulation
Scott Sheffield
Ann. Probab. 44(6): 3804-3848 (November 2016). DOI: 10.1214/15-AOP1061


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 $\mathbb{Z}^{2}$. 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 $p=1/2$, $q=4$.


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Scott Sheffield. "Quantum gravity and inventory accumulation." Ann. Probab. 44 (6) 3804 - 3848, November 2016.


Received: 1 June 2014; Revised: 1 September 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1359.60120
MathSciNet: MR3572324
Digital Object Identifier: 10.1214/15-AOP1061

Primary: 60D05 , 60K35

Keywords: Continuum random tree , FK random cluster model , Liouville quantum gravity , mating of trees , Planar map , Scaling limit , Schramm–Loewner evolution

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 6 • November 2016
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