The TASEP on Galton-Watson trees

TL;DR

This paper analyzes the TASEP on Galton-Watson trees, focusing on convergence to equilibrium, current behavior, and particle trajectory decoupling, providing new insights into particle flow on branching structures.

Contribution

It introduces a detailed study of TASEP on trees, establishing conditions for current behavior and novel bounds on particle trajectory decoupling.

Findings

Convergence of the particle configuration distribution to equilibrium.

Conditions for linear or zero current based on transition rates.

Bounds on the first generation where particle trajectories decouple.

Abstract

We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from $x$ to $y$ at rate $r_{x,y}$ provided $y$ is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time $t$ converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first $n$ particles decouple.