# The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps

## TL;DR

This paper derives the exact phase diagram and steady state properties of a complex, multi-species TASEP with nonlocal boundary jumps, inspired by a Weyl group random walk, extending previous matrix ansatz methods.

## Contribution

It introduces an exact solution for a four-species TASEP with nonlocal boundary conditions, providing explicit formulas for steady states and phase diagrams.

## Key findings

- Exact steady state distribution computed
- Explicit formulas for densities and currents derived
- Complete phase diagram characterized

## Abstract

We consider a finite one-dimensional totally asymmetric simple exclusion process (TASEP) with four types of particles, $\{1,0,\bar{1},*\}$, in contact with reservoirs. Particles of species $0$ can neither enter nor exit the lattice, and those of species $*$ are constrained to lie at the first and last site. Particles of species $1$ enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species $\bar{1}$ enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.