One-dimensional annihilating random walk with long-range interaction

We study the annihilating random walk with long-range interaction in one dimension. Each particle performs random walks on a one-dimensional ring in such a way that the probability of hopping toward the nearest particle is $W= [1 - \epsilon (x+\mu)^{-\sigma}]/2$ (the probability of moving away from its nearest particle is $1-W$), where $x$ is the distance from the hopping particle to its nearest particle and $\epsilon$, $\mu$, and $\sigma$ are parameters. For positive (negative) $\epsilon$, a particle is effectively repulsed (attracted) by its nearest particle and each hopping is generally biased. On encounter, two particles are immediately removed from the system. We first study the survival probability and the mean spreading behaves in the long-time limit if there are only two particles in the beginning. Then, we study how the density decays to zero if all sites are occupied at the outset. We find that the asymptotic behaviors are classified by seven categories: (i) $\sigma>1$ or $\epsilon=0$, (ii) $\sigma = 1$ and $2\epsilon > 1$, (iii) $\sigma=1$ and $2\epsilon = 1$, (iv) $\sigma = 1$ and $2\epsilon < 1$, (v) $\sigma<1$ and $\epsilon > 0$, (vi) $\sigma = 0$ and $\epsilon<0$, and (vii) $0 < \sigma <1$ and $\epsilon<0$. The asymptotic behaviors in each category are universal in the sense that $\mu$ (and sometimes $\epsilon$) cannot affect the asymptotic behaviors.