Front Propagation with Rejuvenation in Flipping Processes

TL;DR

This paper analyzes a stochastic directed flipping process on a 1D lattice, revealing complex front dynamics, depletion zones, and rejuvenation effects through theoretical and numerical methods.

Contribution

It introduces a detailed study of front propagation, depletion zones, and rejuvenation in a directed flipping process relevant to the random edge simplex algorithm.

Findings

Front propagates with a nontrivial velocity.

Depletion zone with excess vacant sites grows logarithmically.

Front rejuvenation varies with age, affecting vigor.

Abstract

We study a directed flipping process that underlies the performance of the random edge simplex algorithm. In this stochastic process, which takes place on a one-dimensional lattice whose sites may be either occupied or vacant, occupied sites become vacant at a constant rate and simultaneously cause all sites to the right to change their state. This random process exhibits rich phenomenology. First, there is a front, defined by the position of the left-most occupied site, that propagates at a nontrivial velocity. Second, the front involves a depletion zone with an excess of vacant sites. The total excess D_k increases logarithmically, D_k ~ ln k, with the distance k from the front. Third, the front exhibits rejuvenation -- young fronts are vigorous but old fronts are sluggish. We investigate these phenomena using a quasi-static approximation, direct solutions of small systems, and numerical simulations.