Eric I. Corwin, Robin Stinchcombe, M. F. Thorpe
Using results for bond percolation on various lattices from 2 to 14 dimensions, we show for the first time that in the limit of large dimension $d$ and/or large number of neighbors $z$, a randomly diluted Erd\"{os}-R\'{e}nyi graph is approached smoothly. This conclusion is reinforced with new results on bond diluted hyper-sphere packs which show how the mean coordination, excess kurtosis and skewness evolve with dimension to the Erd\"{os}-R\'{e}nyi limit. It is demonstrated that this result occurs because loops become irrelevant in percolation in high dimensions.
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http://arxiv.org/abs/1304.3399
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