Matrix-product ansatz for the totally asymmetric simple exclusion process with a generalized update on a ring

We apply the matrix-product ansatz to study the totally asymmetric simple exclusion process on a ring with a generalized discrete-time dynamics depending on two hopping probabilities, $p$ and $\tilde{p}$. The model contains as special cases the TASEP with parallel update, when $\tilde{p}=0$, and with sequential backward-ordered update, when $\tilde{p}=p$. We construct a quadratic algebra and its two-dimensional matrix-product representation to obtain exact finite-size expressions for the partition function, the current of particles, and the two-point correlation function. Our main new result is the derivation of the finite-size pair correlation function. Its behavior is analyzed in different regimes of effective attraction and repulsion between the particles, depending on whether $\tilde{p}>p$ or $\tilde{p}<p$. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation $\tilde{p}\to 1$, when the stationary configurations contain just one cluster.

Figure 1

The dependence on the distance $r$ of the particle-particle correlation function $F_{24,9}^{1,1}(r;\nu )$ on a ring of $L=24$ sites with $N=9$ particles: (a) in the limit $\nu \to 1$, when only configurations containing a single cluster survive (red stars); (b) at $\nu =0$, when the correlations are constant (magenta disks), and (c) at $\nu =-10$, when marked anticorrelations appear at the wings (blue rotated squares).

Figure 2

The dependence on the distance $r$ of the particle-particle correlation function $F_{24,12}^{1,1}(r;\nu )$ on a ring of $L=24$ sites with $N=12$ particles: (a) in the limit $\nu \to 1$, when only single-cluster configurations contribute (red stars), (b) at $\nu =0.5$, when the graph is considerably more smooth and flat (blue rotated squares), and (c) at $\nu =0$, when the correlations are constant (magenta disks).