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 (p) over tilde. The model contains as special cases the TASEP with parallel update, when (p) over tilde = 0, and with sequential backward- ordered update, when (p) over tilde = 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 (p) over tilde > p or (p) over tilde < p. In particular, we explicitly obtain an analytic expression for the pair correlation function in the limit of irreversible aggregation <(p)over tilde> -> 1, when the stationary configurations contain just one cluster.