The Annals of Probability

Quantum gravity and inventory accumulation

Scott Sheffield

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114263

Digital Object Identifier
doi:10.1214/15-AOP1061

Mathematical Reviews number (MathSciNet)
MR3572324

Zentralblatt MATH identifier
1359.60120

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Liouville quantum gravity planar map scaling limit Schramm–Loewner evolution FK random cluster model continuum random tree mating of trees

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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