Discrete-time TASEP with holdback

Abstract

We study the following interacting particle system. There are ρn particles, ρ<1, moving clockwise (“right”), in discrete time, on n sites arranged in a circle. Each site may contain at most one particle. At each time, a particle may move to the right-neighbor site according to the following rules. If its right-neighbor site is occupied by another particle, the particle does not move. If the particle has unoccupied sites (“holes”) as neighbors on both sides, it moves right with probability 1. If the particle has a hole as the right-neighbor and an occupied site as the left-neighbor, it moves right with probability 0<p<1. (We refer to the latter rule as a “holdback” property.) From the point of view of holes moving counter-clockwise, this is a zero-range process.The main question we address is: what is the system steady-state flux (or throughput) when n is large, as a function of density ρ? The most interesting range of densities is 0≤ρ<1∕2. We define the system typical flux as the limit in n→∞ of the steady-state flux in a system subject to additional random perturbations, when the perturbation rate vanishes. Our main results show that: (a) the typical flux is different from the formal flux, defined as the limit in n→∞ of the steady-state flux in the system without perturbations, and (b) there is a phase transition at density h=p∕(1+p). If ρ<h, the typical flux is equal to ρ, which coincides with the formal flux. If ρ>h, a condensation phenomenon occurs, namely the formation and persistence of large particle clusters; in particular, the typical flux in this case is p(1−ρ)<h<ρ, which differs from the formal flux when h<ρ<1∕2.Our results include both the steady-state analysis (which determines the typical flux) and the transient analysis. In particular, we derive a version of the Ballot Theorem, and show that the key “reason” for large cluster formation for densities ρ>h is described by this theorem.