Absorbing-state phase transitions with extremal dynamics

TL;DR

This paper explores extremal dynamics in phase transitions to absorbing states, linking extremal and nonextremal processes, and introduces an extremal directed percolation universality class with distinct critical behavior.

Contribution

It defines an extremal absorbing process, relates it to extremal and nonextremal models, and characterizes a new universality class with modified critical exponents and behavior.

Findings

Refined estimates for the Bak-Sneppen model's threshold and exponents.

Identification of an extremal directed percolation universality class.

Demonstration that asymmetric updating affects the extremal class significantly.

Abstract

Extremal dynamics represents a path to self-organized criticality in which the order parameter is tuned to a value of zero. The order parameter is associated with a phase transition to an absorbing state. Given a process that exhibits a phase transition to an absorbing state, we define an ``extremal absorbing" process, providing the link to the associated extremal (nonabsorbing) process. Stationary properties of the latter correspond to those at the absorbing-state phase transition in the former. Studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible. In the case of the Bak-Sneppen (BS) model, the absorbing version is closely related to the "$f$-avalanche" introduced by Paczuski, Maslov and Bak [Phys. Rev. E {\bf 53}, 414 (1996)], or, in spreading simulations to the "BS branching process" also studied by these authors. The corresponding nonextremal process belongs to the directed percolation universality class. We revisit the absorbing BS model, obtaining refined estimates for the threshold and critical exponents in one dimension. We also study an extremal version of the usual contact process, using mean-field theory and simulation. The extremal condition slows the spread of activity and modifies the critical behavior radically, defining an ``extremal directed percolation" universality class of absorbing-state phase transitions. Asymmetric updating is a relevant perturbation for this class, even though it is irrelevant for the corresponding nonextremal class.