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

TL;DR

This paper investigates phase transitions in a one-dimensional branching annihilating random walk with long-range repulsion, revealing logarithmic scaling, reentrant phase transitions, and crossover behaviors depending on interaction parameters.

Contribution

It introduces a detailed analysis of how long-range repulsive interactions influence phase transitions and reentrant behaviors in branching annihilating random walks.

Findings

Reentrant phase transitions occur for odd offspring number when interaction exceeds a threshold.

System with even offspring number remains active for nonzero branching rate if interaction exceeds the threshold.

Crossover exponent for attractive interaction with even offspring number is approximately 1.123.

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 $\varepsilon x^{-\sigma}$, where $x$ is the distance from a particle to its closest particle, $0\le \sigma \le 1$, and the sign of $\varepsilon$ determines whether the interaction is repulsive (positive $\varepsilon$) or attractive (negative $\varepsilon$). A state without particles is the absorbing state. We find a threshold $\varepsilon_s$ such that the absorbing state is dynamically stable for small branching rate $q$ if $\varepsilon < \varepsilon_s$. The threshold differs significantly, depending on parity of the number $\ell$ of offspring. When $\varepsilon>\varepsilon_s$, 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 $\varepsilon>\varepsilon_s$. 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 $0<\varepsilon < \varepsilon_s$. We also study the crossover behavior for $\ell = 2$ when the interaction is attractive (negative $\varepsilon$), to find the crossover exponent $\phi=1.123(13)$ for $\sigma=0$.