Degenerate Competing Three-Particle Systems

TL;DR

This paper investigates three-particle systems with rank-dependent drifts and variances, focusing on degenerate cases where variances vanish, leading to ballistic motion and diverse behaviors, along with stability analysis of the gap process.

Contribution

It introduces and analyzes degenerate three-particle systems with rank-based parameters, highlighting the effects of vanishing variances on system dynamics and stability.

Findings

Degenerate systems exhibit ballistic behavior when variances vanish.

Different rank configurations lead to markedly different system dynamics.

Stability properties of the gap process are characterized.

Abstract

We study systems of three interacting particles, in which drifts and variances are assigned by rank. These systems are "degenerate": the variances corresponding to one or two ranks can vanish, so the corresponding ranked motions become ballistic rather than diffusive. Depending on which ranks are allowed to "go ballistic" the systems exhibit markedly different behavior which we study in some detail. Also studied are stability properties for the resulting planar process of gaps between successive ranks.