The effect of quenched disorder in neutral theories

We study systems with two symmetric absorbing states, such as the voter model and its variations, which have been broadly used as minimal neutral models in genetics, population ecology, sociology, etc. We analyze the effects of a key ingredient inevitably present in most real applications: random-field-like quenched disorder. In accord with simulations and previous findings, coexistence between the two competing states/opinions turns out to be strongly favored by disorder in the standard voter model; actually, a disorder-induced phase transition is generated for any finite system size in the presence of an arbitrarily small spontaneous-inversion rate (preventing absorbing states from being stable). For nonlinear versions of the voter model a general theory (by Al Hammal et al) explains that the spontaneous breaking of the up/down symmetry and an absorbing state phase transition can occur either together or separately, giving rise to two different scenarios. Here, we show that the presence of quenched disorder in nonlinear voter models does not allow the separation of the up down (Ising-like) symmetry breaking from the active-to-absorbing phase transition in low-dimensional systems: both phenomena can occur only simultaneously, as a consequence of the well-known Imry-Ma argument generalized to these non-equilibrium problems. When the two phenomena occur in unison, resulting in a genuinely non-equilibrium (`generalized voter') transition, the Imry Ma argument is violated and the symmetry can be spontaneously broken even in low dimensions.