The phase transition in bounded-size Achlioptas processes

Abstract

Perhaps the best understood phase transition is that in the component structure of the uniform random graph process introduced by Erdős and Rényi around 1960. Since the model is so fundamental, it is very interesting to know which features of this phase transition are specific to the model, and which are `universal', at least within some larger class of processes (a `universality class'). Achlioptas process, a class of variants of the Erdős--Rényi process that are easy to define but difficult to analyze, have been extensively studied from this point of view. Here, settling a number of conjectures and open problems, we show that all `bounded-size' Achlioptas processes share (in a strong sense) all the key features of the Erdős--Rényi phase transition. We do not expect this to hold for Achlioptas processes in general.

97 pages, 7 figures; to appear in Memoirs of the AMS

References in corpus (6)

• "Explosive Percolation" Transition is Actually Continuous
• Random subgraphs of finite graphs: I. The scaling window under the triangle condition
• Achlioptas process phase transitions are continuous
• Achlioptas processes are not always self-averaging
• The evolution of subcritical Achlioptas processes
• On the critical probability in percolation

Cited by in corpus (5)

• On Wormald's differential equation method
• On the critical probability in percolation
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