  We consider an exclusion process on a ring in which a particle hops to an
empty neighbouring 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.
