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Effect of diffusion in one-dimensional discontinuous absorbing phase transitions

It is known that diffusion provokes substantial changes in continuous absorbing phase transitions. Conversely, its effect on discontinuous transitions is much less understood. In order to shed light in this direction, we study the inclusion of diffusion in the simplest one-dimensional model with a discontinuous absorbing phase transition, namely the long-range contact process ($σ$-CP). Particles interact as in the usual CP, but the transition rate depends on the length $\ell$ of inactive sites according to $1 + a \ell^{-σ}$, where $a$ and $σ$ are control parameters. In the absence of diffusion, this system presents both a discontinuous and a continuous phase transition, depending on the value of $σ$. The inclusion of diffusion in this model has been investigated by numerical simulations and mean-field calculations. Results show that there exists three distinct regimes. For sufficiently low and large $σ$'s the transition is respectively always discontinuous or continuous, independently of the strength of the diffusion. On the other hand, in an intermediate range of $σ$'s, the diffusion causes a suppression of the phase coexistence leading to a continuous transition belonging to the DP universality class. This set of results does not agree with mean-field predictions, whose reasons will be discussed further.