Quantum gravity and inventory accumulation

TL;DR

This paper connects inventory accumulation models with random planar maps and Liouville quantum gravity, showing how certain random walks scale to Brownian motion and how these models converge to continuous random surfaces.

Contribution

It establishes a novel link between inventory accumulation processes and the scaling limits of random planar maps, advancing understanding of their convergence to quantum gravity surfaces.

Findings

Random walks in inventory models scale to Brownian motions with p-dependent diffusion matrices.

A bijection between inventory trajectories and decorated planar maps enables convergence results.

Phase transition identified at p=1/2, q=4 in the models.

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 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.