Hipster random walks

TL;DR

This paper introduces 'hipster random walks', a new family of processes on trees, and establishes their distributional convergence by linking them to convection-diffusion equations and numerical PDE analysis.

Contribution

It defines hipster random walks and proves their convergence using PDE analogies and numerical analysis techniques, connecting to prior models on trees and hierarchical lattices.

Findings

Distributional convergence of hipster random walks

Connection to convection-diffusion equations

Application of numerical PDE convergence results

Abstract

We introduce and study a family of random processes on trees we call hipster random walks, special instances of which we heuristically connect to the min-plus binary trees introduced by Robin Pemantle and studied by Auffinger and Cable (2017; arXiv:1709.07849), and to the critical random hierarchical lattice studied by Hambly and Jordan (2004). We prove distributional convergence for the processes by showing that their evolutions can be understood as a discrete analogues of certain convection-diffusion equations, then using a combination of coupling arguments and results from the numerical analysis literature on convergence of numerical approximations of PDEs.