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Fractal Geometry of Spin-Glass Models


  • J. F. Fontanari
  • P. F. Stadler


Stability and diversity are two key properties that living entities share with spin glasses, where they are manifested through the breaking of the phase space into many valleys or local minima connected by saddle points. The topology of the phase space can be conveniently condensed into a tree structure, akin to the biological phylogenetic trees, whose tips are the local minima and internal nodes are the lowest-energy saddles connecting those minima. For the infinite-range Ising spin glass with p-spin interactions, we show that the average size-frequency distribution of saddles obeys a power law $\sim$ w-D, where w=w(s) is the number of minima that can be connected through saddle s, and D is the fractal dimension of the phase space.

Suggested Citation

  • J. F. Fontanari & P. F. Stadler, 2001. "Fractal Geometry of Spin-Glass Models," Working Papers 01-06-034, Santa Fe Institute.
  • Handle: RePEc:wop:safiwp:01-06-034

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    References listed on IDEAS

    1. Martin Shubik, 2000. "The Theory of Money," Working Papers 00-03-021, Santa Fe Institute.
    2. Arthur, W Brian, 1989. "Competing Technologies, Increasing Returns, and Lock-In by Historical Events," Economic Journal, Royal Economic Society, vol. 99(394), pages 116-131, March.
    3. Kiyotaki, Nobuhiro & Wright, Randall, 1989. "On Money as a Medium of Exchange," Journal of Political Economy, University of Chicago Press, vol. 97(4), pages 927-954, August.
    4. Dubey, Pradeep & Mas-Colell, Andreau & Shubik, Martin, 1980. "Efficiency properties of strategies market games: An axiomatic approach," Journal of Economic Theory, Elsevier, vol. 22(2), pages 339-362, April.
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    Spin glass; landscape; fractal; barrier tree;

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