Two-point correlation function of an exclusion process with hole-dependent rates

We consider an exclusion process on a ring in which a particle hops to an empty neighboring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well-known mapping of this model to the zero-range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n) = 1 + (b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b > 2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b - 2. We also calculate the two-point correlation function above the critical density and find that the correlation length diverges with a critical exponent nu = 1/(b - 2) for b < 3 and 1 for b > 3. These results are compared with those obtained using an exact series expansion for finite systems.