Analytical results for stochastically growing networks: connection to the zero-range process

Cluster-factorized steady states in finite-range processes

We study a class of nonequilibrium lattice models on a ring where particles hop in a particular direction, from a site to one of its (say, right) nearest neighbors, with a rate that depends on the occupation of all the neighboring sites within a range R. This finite-range process (FRP) for R=0 reduces to the well-known zero-range process (ZRP), giving rise to a factorized steady state (FSS) for any arbitrary hop rate. We show that, provided the hop rates satisfy a specific condition, the steady state of FRP can be written as a product of a cluster-weight function of (R+1) occupation variables. We show that, for a large class of cluster-weight functions, the cluster-factorized steady state admits a finite dimensional transfer-matrix formulation, which helps in calculating the spatial correlation functions and subsystem mass distributions exactly. We also discuss a criterion for which the FRP undergoes a condensation transition.

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 ν=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.