Role of loop activity in correlated percolation: results on a Husimi cactus and relationship with a Bethe lattice

We consider the problem of correlated percolation on a Husimi cactus, which allows finite loops of size l, to investigate the effects of loop formation on percolation properties. In particular, we calculate how the percolation threshold and the percolation probability depend on l and the loop activity n. We calculate the contribution and its dependence on I and n from finite and infinite clusters to all densities. We show that macroscopic loops are formed immediately after percolation, and we calculate their density dependence on I and n. We compare the results on Husimi cactus with those on a Bethe lattice. We finally establish that the Husimi cactus turns into a Bethe lattice as l --> infinity.