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Analogs of renormalization group transformations in random processes

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  • Shlesinger, Michael F.
  • Hughes, Barry D.

Abstract

We review the properties of a real-space renormalization group transformation of the free energy, including the existence of oscillatory terms multiplying the non-analytic part of the free energy. We then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation. (The essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary.) These random processes are shown to be generated directly from Cantor sets. In a spatial representation, the ensuing random process exhibits a transition between Gaussian and fractal behavior. In the fractal regime, the trajectories will, in an average sense, form self-similar clusters. In a temporal representation, the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior. The fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses.

Suggested Citation

  • Shlesinger, Michael F. & Hughes, Barry D., 1981. "Analogs of renormalization group transformations in random processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 109(3), pages 597-608.
    • Handle: RePEc:eee:phsmap:v:109:y:1981:i:3:p:597-608
    DOI: 10.1016/0378-4371(81)90015-7
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    Cited by:

    1. Vlad, Marcel Ovidiu & Mackey, Michael C., 1996. "Industrial replacement, communication networks and fractal time statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 229(3), pages 295-311.

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