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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 ρ of particles, from a low-density phase with all particles immobile for ρ ≤ ρc = 1 n+1, to an active state for ρ > ρc. The mean fraction of movable particles in the active steady state varies as (ρ − ρc) β , for ρ near ρc. We show that for the model with range n, the exponent β = n, and thus can be made arbitrarily large. Introduction -There has recently been a lot of interest in models showing an active-absorbing state transition [1][2][3]. The best studied of these is the directed percolation (DP) class [4,5], and the related pair-contact process with diffusion (PCPD) [6][7][8] and parity-conserving directed percolation (PC) [9,10]. Models with conserved number of particles where a particle may hop only if there are sufficient number of other particles present within a given range, also show an active-absorbing phase transition, where the lowdensity phase is inactive. Examples of this class are the conserved lattice gas model and the conserved threshold transfer process[11], fixed energy sandpiles [12] and the activated random walkers model [13]. By adding a slow drive and dissipation, these models can be converted into models showing self-organized criticality [14]. It has recently been argued that on adding symmetry-breaking perturbations, these models would flow to DP-like critical behavior, but the evidence for this scenario is mainly numerical [15,16]. It seems worthwhile to investigate further models with conserved particle number showing active-absorbing phase transitions.
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