A class of exactly solved assisted hopping models of active-absorbing state transitions on a line

We construct a class of assisted hopping models in one dimension in which a particle can move only if it does not lie in an otherwise empty interval of length greater than $n+1$. We determine the exact steady state by a mapping to a gas of defects with only on-site interaction. We show that this system undergoes a phase transition as a function of the density $\rho$ of particles, from a low-density phase with all particles immobile for $\rho \le \rho_c = \frac{1}{n+1}$, to an active state for $\rho > \rho_c$. The mean fraction of movable particles in the active steady state varies as $(\rho - \rho_c)^{\beta}$, for $\rho$ near $\rho_c$. We show that for the model with range $n$, the exponent $\beta =n$, and thus can be made arbitrarily large.