# Discrete-time TASEP with holdback

## TL;DR

This paper analyzes a discrete-time particle system with holdback rules on a circle, revealing a phase transition at a critical density where large clusters form, affecting the system's throughput.

## Contribution

It introduces a new model with holdback interactions, characterizes the phase transition, and distinguishes between typical and formal flux in large systems.

## Key findings

- Typical flux differs from formal flux due to perturbations.
- A phase transition occurs at density h=p/(1+p).
- Large particle clusters form when density exceeds h.

## Abstract

We study the following interacting particle system. There are $\rho n$ particles, $\rho < 1$, 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.) The main question we address is: what is the system steady-state flux (or throughput) when $n$ is large, as a function of density $\rho$? The most interesting range of densities is $0\le \rho < 1/2$. We define the system {\em typical flux} as the limit in $n\to\infty$ 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 $n\to\infty$ of the steady-state flux in the system without perturbations, and (b) there is a phase transition at density $h=p/(1+p)$. If $\rho<h$, the typical flux is equal to $\rho$, which coincides with the formal flux. If $\rho>h$, a {\em condensation} phenomenon occurs, namely the formation and persistence of large particle clusters; in particular, the typical flux in this case is $p(1-\rho) < h < \rho$, which differs from the formal flux when $h < \rho < 1/2$. Our results include both steady-state and transient analysis. In particular, we derive a version of the Ballot Theorem, and show that the key "reason" for large cluster formation for densities $\rho > h$ is described by this theorem.