Abstract
The cluster density function of independent percolationκ in ad-dimensional lattice is considered. For eachn, it is shown thatκ(p) has finitenth leftderivative at critical probabilityp c ifd is sufficiently large. This result agrees with the Bethe lattice approximation, where thenth one-sided derivative ofκ(p) is bounded atp c for alln.
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Yang, W.S., Zhang, Y. A note on differentiability of the cluster density for independent percolation in high dimensions. J Stat Phys 66, 1123–1138 (1992). https://doi.org/10.1007/BF01055720
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DOI: https://doi.org/10.1007/BF01055720