Behavior dominated by slow particles in a disordered asymmetric exclusion process. (Q1879921)

This paper studies large space and large time scales below and until the hydrodynamic scale of totally asymmetric simple exclusion process (TASEP) with random rates. The density of (slow) particles is known to be an order parameter of a phase transition that could occur when the distribution of the rate has a large enough appropriate tail. TASEP is related to similar probabilistic behaviors modelling simple models of single lane traffic, and is also linked to the zero-range process which represents the gaps between vehicles. Each vehicle (or particle) carries its own Poisson clock at a random rate \(p_i\) and whenever the clock rings, the process advances one step to the right provided the next site is vacant. Any invariant measure should be of a product from where each particle motion is marginally a Poisson process with a certain rate \(a\). The exclusion rule implies that fast particles cannot overpass slow particles and this creates large clusters of particles trapped behind unusually slow particles. When a large cluster meets another one, the motion slows down even more and at low density, there is a rigid transformation of the initial profile without any equilibrium distribution. This paper focuses on the speed of this slow-down. It appears at a scale below the hydrodynamic scale, where it is immediate because hydrodynamic limit reveals only the trivial final behavior with a macroscopically constant low density profile. The authors finds the correct order of correction from the hydrodynamic limit for annealed distributions, i.e. with averaged random rates, and studies a partially related question mimicking an outflow jam when all particles start packed to the left. The proofs rely on several couplings with different rates and different starting points.