Characterize independent-set behavior for fixed percolation parameters below one

Characterize the behavior of the number of independent sets in the percolated hypercube for every fixed percolation parameter p<1, including the regimes not resolved by the existing central-limit and lognormal-limit results.

Background

The paper reviews prior work on the number of independent sets in a randomly percolated hypercube. Earlier results established distributional limits in certain parameter regimes, including parameters tending to one, but did not determine the behavior for an arbitrary fixed p<1. The unresolved issue concerns the asymptotic behavior of the random partition function, or equivalently the total number of independent sets, across the remaining fixed-parameter regimes.