Branching annihilating random walk with long-range repulsion: logarithmic scaling, reentrant phase transitions, and crossover behaviors

Abstract

We study absorbing phase transitions in the one-dimensional branching annihilating random walk with long-range repulsion. The repulsion is implemented as hopping bias in such a way that a particle is more likely to hop away from its closest particle. The bias strength due to long-range interaction has the form $\epsilon x^{-\sigma }$, where $x$ is the distance from a particle to its closest particle, $0\le \sigma \le 1$, and the sign of $\epsilon$ determines whether the interaction is repulsive (positive $\epsilon$) or attractive (negative $\epsilon$). A state without particles is the absorbing state. We find a threshold ${\epsilon }_s$ such that the absorbing state is dynamically stable for small branching rate $q$ if $\epsilon <{\epsilon }_s$. The threshold differs significantly, depending on parity of the number $\ell$ of offspring. When $\epsilon >{\epsilon }_s$, the system with odd $\ell$ can exhibit reentrant phase transitions from the active phase with nonzero steady-state density to the absorbing phase, and back to the active phase. On the other hand, the system with even $\ell$ is in the active phase for nonzero $q$ if $\epsilon >{\epsilon }_s$. Still, there are reentrant phase transitions for $\ell =2$. Unlike the case of odd $\ell$, however, the reentrant phase transitions can occur only for $\sigma =1$ and $0<\epsilon <{\epsilon }_s$. We also study the crossover behavior for $\ell =2$ when the interaction is attractive (negative $\epsilon$), to find the crossover exponent $\varphi =1.123(13)$ for $\sigma =0$.

10 pages, 8 figures

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