Asymmetric cooperative motion in one dimension

TL;DR

This paper proves distributional convergence for asymmetric cooperative motions on the integer lattice, generalizing previous models and introducing a novel approach connecting recurrence relations to Hamilton-Jacobi equations.

Contribution

It introduces a new class of asymmetric cooperative motion processes and a novel analytical method linking recurrence relations to Hamilton-Jacobi theory.

Findings

Distributional convergence of the processes is established.

Surprising lattice effects are observed in the distributional limits.

The approach connects recurrence relations with finite difference schemes for PDEs.

Abstract

We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which we call asymmetric cooperative motions. The model generalizes the "totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point out some surprising lattice effects that can persist in the distributional limit, and propose several generalizations and directions for future research.