Quantum gravity and inventory accumulation
type: Item
instance of: scholarly article
title: Quantum gravity and inventory accumulation
zbMATH Open document ID: 1359.60120
DOI: 10.1214/15-AOP1061
author: Scott Sheffield
published in: The Annals of Probability
publication date: 2017-01-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.2241
review text: The paper under review studies inventory accumulation at a LIFO (last-in-first-out) retailer with two products and shows that the corresponding random walks scale to Brownian motions with diffusion matrices depending on the fraction \(p\) of customers ordering the ``freshest available'' product type (Theorem 2.5). The author starts with stating and proving a theorem about inventory accumulation (Sections 2 and 3) and finds a surprising phase transition: there is no macroscopic inventory imbalance when \(p\geq 1/2\). Using two products, say hamburger and cheeseburger, one can create a set of alphabets \(\Theta =\{a, b, c, d, e\}\) with \(a=\) cheeseburger, \(b=\) hamburger, \(c=\) cheeseburger order, \(d=\) hamburger order, \(e=\) flexible order. A word from \(\Theta\) represents two products produced in a day with natural orders. For every finite-length word in \(\Theta\), up to commutativity \(ad=da, bc=cb\) and order fulfillment relations \(ac=bd=ae=be=\emptyset\), there is a unique reduced word in \({\mathcal G}\). Let \(\mu\) be the corresponding probability measure on the space \(\Omega\) of maps from \(\mathbb Z\) to \(\Theta\). One writes \(m=\phi (n)\) and \(n=\phi (m)\) if \(m\) and \(n\) are matched, i.e., a burger added at time \(m\) is consumed at time \(n\). Every \(X(j)\), representing a day in the life of the restaurant, has a unique match almost surely so that \(\phi\) is an involution on \(\mathbb Z\) (Proposition 2. 2). The main result of this paper is Theorem 2.5, which states that the scaling random function \(\varepsilon A_{t/\varepsilon^2}\) converges in law with respect to the \(L^{\infty}\)-metric on any compact interval to two independent standard one-dimensional Brownian motions \((B^1_{\alpha t}, B^1_t)\) with \(\alpha = \max \{1- 2p, 0\}\), where the random process \(A_n = (D_n, C_n)\) keeps track of the net change in the burger count and the burger discrepancy. The net inventory accumulation between time zero and time \(n\) is close to b…
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J67
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 82B41
zbMATH DE Number: 6674838
zbMATH Keywords: inventory accumulation
zbMATH Keywords: Liouville quantum gravity
zbMATH Keywords: planar map
zbMATH Keywords: scaling limit
zbMATH Keywords: random walks
zbMATH Keywords: Brownian motion
zbMATH Keywords: Schramm-Loewner evolution
zbMATH Keywords: FK random cluster model
zbMATH Keywords: continuum random tree
MaRDI profile type: Publication
arXiv ID: 1108.2241
OpenAlex ID: W2963346680
description: scientific article