Abstract
A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponentsβ=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp c (m=1)=l/(z − 1) andp c(m=z) =[l/(z − 1)]1/(z−1). The cluster size distribution asymptotically decays exponentially withn, for largen, p ≠ p c .
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Supported in part by National Science Foundation grant DMR78-10813.
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Reich, G.R., Leath, P.L. High-density percolation: Exact solution on a Bethe lattice. J Stat Phys 19, 611–622 (1978). https://doi.org/10.1007/BF01011772
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DOI: https://doi.org/10.1007/BF01011772