Fixation in cyclically competing species on a directed graph with   quenched disorder
  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 $c_l(s)$. 
Hiroki Ohta, Namiko Mitarai
cond-mat.stat-mech cond-mat.dis-nn