Critical behavior of models with infinitely many absorbing states

Abstract

In this article we review several models with many absorbing con gurations. In all these models attention will be focused on the in uence of the initial state on the dynamic evolution. The relation of systems with many absorbing con gurations to those displaying avalanches self-organized criticality is also investigated. Some new results are presented both for systems with and without parity conservation. Recently derived scaling relations are tested. I IntroductionThe study of nonequilibrium models exhibiting phase transitions from an active to an absorbing phase has been a topic of intense research in the last years 1-11 since Broadbent and Hammersley 12 i n troduced directed percolation DP 13-15 . Later studies have connected this to other elds, and the topic has become familiar in many processes encountered in nature. Among them we refer to models of epidemics 19-21 , catalytic kinetics 22 , surface growth 23 , and selforganized criticality 24 . Most of these models were found to belong to the universality class of DP, supporting the conjecture of Jenssen 25 and Grassberger 26 that all systems exhibiting a continuous transition into a unique absorbing state, without any other symmetry or conservation law, belong to the DP universality class. This conjecture was later extended by Grinstein et al. 27 to systems with multiple components. Exceptions to this universality class were introduced by Grassberger 28,29 ; they correspond to models with doubly degenerated absorbing states and modulo 2 parity conservation.More recently Jensen 30 i n troduced a model the pair contact process -PCP that is the prototype of a model with a number of absorbing con gurations growing with the system size, which goes to in nity in the thermodynamic limit. Other models with innitely many absorbing states IMAS were introduced since then, and have proved to have a more complex dynamics than the ones with only a single absorbing state. Apart from this, all them of show -in agreement with the previous conjecture -a DP-like behavior in what concerns the static critical behavior. However, the critical exponents that govern the spreading from an initial seed the closest con guration to the absorbing one show non-universal behavior. The critical exponents describing time dependence depend on the initial con guration of the system. Only for the "natural" conguration the one obtained from the evolution of the system are DP values recovered. An attempt to justify this behavior on solid grounds was made by M ũ noz et al. 31 using a continuous eld theory. Other features of IMAS models were discovered recently by Lipowski 32 , who observed that a model inspiried by biological evolution" in 2+1-dimensions has the critical behavior of 1+1-dimensional DP an e ect similar to the dimensional reduction encountered in random systems 33 .Another topic of intense research in our days is selforganized criticality SOC. Recently some attempts to connect SOC with absorbing-state phase transitions were made 34, 35 . SOC was introduced by Bak, Tang and Weisenf...

Non-universal spreading exponents in a catalytic reaction model

“…In their analysis, they considered exponents that can change with the initial condition, leading to a proper manner for relating these dynamic critical exponents. Their approach was initially applied in the studies of the threshold transfer process [31] and of the dimer reaction model [29], being later considered for other different contexts [32]- [34].…”