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Achlioptas processes are not always self-averaging.

Oliver Riordan1, Lutz Warnke

  • 1Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary
This summary is machine-generated.

Achlioptas processes, a class of percolation models, challenge the assumption of self-averaging. Some models exhibit persistent random fluctuations in their order parameter, even in the thermodynamic limit.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Network Science

Background:

  • Achlioptas processes are a class of percolation models studied for their unique properties.
  • Research has primarily focused on the order parameter, representing the largest connected component size.
  • Classical percolation models typically exhibit self-averaging behavior.

Purpose of the Study:

  • To investigate the self-averaging property in Achlioptas processes.
  • To determine if the order parameter in these models universally self-averages.
  • To identify conditions under which self-averaging breaks down.

Main Methods:

  • Analysis of a class of Achlioptas processes.
  • Examination of the order parameter's behavior.
  • Study of system-size dependence and thermodynamic limit.

Main Results:

  • Demonstrated that self-averaging is not a universal feature of Achlioptas processes.
  • Identified specific Achlioptas processes where the order parameter exhibits significant random fluctuations.
  • These fluctuations persist even as the system size approaches the thermodynamic limit.

Conclusions:

  • The breakdown of self-averaging is a notable characteristic of certain Achlioptas processes.
  • This finding contrasts with the behavior observed in classical percolation models.
  • Future research should consider the implications of non-self-averaging in these complex systems.