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Self-avoiding walks in quenched random environments

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Abstract

The self-avoiding walk in a quenched random environment is studied using real-space and field-theoretic renormalization and “Flory” arguments. These methods indicate that the system is described, ford<d c =4, and, for large disorder ford>d c , by a strong disorder fixed point corresponding to a “glass” state in which the polymer is confined to the lowest energy path. This fixed point is characterized by scaling laws for the size of the walk,L∼N pζ withN the number of steps, and the fluctuations in the free energy,Αf∼L pζ. The bound 1/ζ-ω⩽d/2 is obtained. Exact results on hierarchical lattices yieldζ>ζ pure and suggests that this inequality holds ford=2 and 3, althoughζ=ζ pure cannot be excluded, particularly ford=2. Ford>d c there is a transition between strong and weak disorder phases at whichζ=ζ pure. The strong-disorder fixed point for SAWs on percolation clusters is discussed. The analogy with directed walks is emphasized.

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Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, Massachusetts

    P. Le Doussal

  2. Department of Physics and Astronomy, University of Massachusetts, 01003, Amherst, Massachusetts

    J. Machta

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  1. P. Le Doussal
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  2. J. Machta
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Le Doussal, P., Machta, J. Self-avoiding walks in quenched random environments. J Stat Phys 64, 541–578 (1991). https://doi.org/10.1007/BF01048306

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  • DOI: https://doi.org/10.1007/BF01048306

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