Central Limit Theorem for a Class of Linear Systems

We consider a class of interacting particle systems with values in $[0,\text{8}{)}^{\text{zd}}$, of which the binary contact path process is an example. For $d \ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.