Fixation in cyclically competing species on a directed graph with quenched disorder

Hiroki Ohta; Namiko Mitarai

Fixation in cyclically competing species on a directed graph with quenched disorder

Statistical Mechanics 2014-02-27 v1 Disordered Systems and Neural Networks

Authors: Hiroki Ohta , Namiko Mitarai

Abstract

A simple model of cyclically competing species on a directed graph with quenched disorder is proposed as an extension of the rock-paper-scissors model. By assuming that the effects of loops in a directed random graph can be ignored in the thermodynamic limit, it is proved for any finite disorder that the system fixates to a frozen configuration when the species number $s$ is larger than the spatial connectivity $c$ , and otherwise stays active. Nontrivial lower and upper bounds for the persistence probability of a site never changing its state are also analytically computed. The obtained bounds and numerical simulations support the existence of a phase transition as a function of disorder for $1<c_l(s)\le c <s$ , with a $s$ -dependent threshold of the connectivity $c_l(s)$ .

Keywords

cyclically competing species rock-paper-scissors model directed graph quenched disorder fixation phase transition persistence probability

Cite

@article{arxiv.1402.6624,
  title  = {Fixation in cyclically competing species on a directed graph with quenched disorder},
  author = {Hiroki Ohta and Namiko Mitarai},
  journal= {arXiv preprint arXiv:1402.6624},
  year   = {2014}
}

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15 pages, 11 figures

Fixation in cyclically competing species on a directed graph with quenched disorder

Abstract

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