Quantum gravity and inventory accumulation
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 being evenly divided between hamburger and cheeseburger with high probability when \(n\) is large, by the main Theorem 2.5. A LIFO retailer accumulates major inventory discrepancies if and only if more than half of its customers have a product preference. Section 3 presents a discrete and elementary proof of Theorem 2.5 relying on standard results in optimal stopping theory. Starting with variance calculations and two simple finite expectation criteria, the infimum over the set of positive numbers is shown to be a.s. finite with finite expectation (Proposition 3. 8). The proof of Theorem 2.5 is given in subsection 3.6, following from various technical lemmas. Section 4 introduces a bijection between inventory accumulation trajectories and possible instances of critical Fortuin-Kasteleyn (FK) random planar maps. The inventory-trajectory central limit theorem becomes a scaling limit theorem about the corresponding loop-decorated surfaces. Table 2 presents correspondences between decorated maps and words. The random map \(X(m, n-m): \mathbb Z \to \Theta\) has a probability bounded below (independent of \(n\)) of containing at most \(m\) elements by Theorem 2.5 with \(m=[\sqrt{n}]\). Given a sequence \(X(\cdot )\) sampled from \(\mu\), the corresponding sequence \(Y(\cdot )\) can be used to construct an infinite random surface with an infinite spanning tree and spanning dual tree. In this manner, Theorem 2.5 can be understood as a scaling limit result about this random pair of infinite trees. The appendix introduces the problem of relating discrete random surfaces to continuum objects like Liouville quantum gravity, conformal loop ensembles and the Schramm-Loewner evolution, to outline a larger project that motivates this paper. It would be interesting to see if this work can be extended to inventory accumulation at a LIFO retailer with three products or more generally \(p\) products.
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