## The Annals of Probability

### Quantum gravity and inventory accumulation

Scott Sheffield

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

#### Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3804-3848.

Dates
Revised: September 2015
First available in Project Euclid: 14 November 2016

https://projecteuclid.org/euclid.aop/1479114263

Digital Object Identifier
doi:10.1214/15-AOP1061

Mathematical Reviews number (MathSciNet)
MR3572324

Zentralblatt MATH identifier
1359.60120

#### Citation

Sheffield, Scott. Quantum gravity and inventory accumulation. Ann. Probab. 44 (2016), no. 6, 3804--3848. doi:10.1214/15-AOP1061. https://projecteuclid.org/euclid.aop/1479114263

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