Abstract
It is rigorously proved that the analog of the free energy for the bond and site percolation problem on\(\mathbb{Z}^v \) in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.
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Supported by the Fonds National Suisse de la Recherche Scientifique (to HK).
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Kunz, H., Souillard, B. Essential singularity in percolation problems and asymptotic behavior of cluster size distribution. J Stat Phys 19, 77–106 (1978). https://doi.org/10.1007/BF01020335
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DOI: https://doi.org/10.1007/BF01020335