Abstract
This article reviews some effects of disorder in percolation systems away from the critical density p c. For densities below p c, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee-Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singularities in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biased diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field.
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This definition differs from the definition often used in the literature, where ϑ is usually defined in terms of the number of unrooted animals. As the number of unrooted clusters of n sites is just A n/n, this definition gives a value 1 larger than the one used by us. For a recent treatment of this result, see D C Brydges and J Z Imbrie, preprint mathphy/0107005.
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Dhar, D. Percolation systems away from the critical point. Pramana - J Phys 58, 419–426 (2002). https://doi.org/10.1007/s12043-002-0025-x
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DOI: https://doi.org/10.1007/s12043-002-0025-x