Minimal path on the hierarchical diamond lattice


We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Lévy's distributions with a power-law decay at-∞, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.

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Roux, S., Hansen, A., da Silva, L.R. et al. Minimal path on the hierarchical diamond lattice. J Stat Phys 65, 183–204 (1991).

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Key words

  • Minimal path
  • directed polymer
  • hierarchical lattice
  • percolation