The TASEP on Galton–Watson trees

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