Branching annihilating random walks with long-range attraction in one dimension

We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest particle has larger transition rate than hopping against the direction. Still, unlike the Lévy flight, a particle only hops to one of its nearest-neighbor sites. The strength of bias takes the form $x^{-σ}$ with non-negative $σ$, where $x$ is the distance to the nearest particle from a particle to hop. By extensive Monte Carlo simulations, we show that the critical decay exponent $δ$ varies continuously with $σ$ up to $σ=1$ and $δ$ is the same as the critical decay exponent of the directed Ising (DI) universality class for $σ\ge 1$. Investigating the behavior of the density in the absorbing phase, we argue that $σ=1$ is indeed the threshold that separates the DI and non-DI critical behavior. We also show by Monte Carlo simulations that branching bias with symmetric hopping exhibits the same critical behavior as the BAWL.

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