Graphs for Menshikov-Zuev's Problems on ρ-percolation Model

In 1993, Menshikov and Zuev introduced ρ−percolation model, in which a path of a graph is ρ−passable in a bond percolation configuration if the concentration of open bonds on it is at least ρ, and concerning this model, they gave four open problems. In this paper, we answer three problems out of them : the first one is whether the ρ−percolation critical probability is equal to the critical probability corresponding to finite/infinite expectation of the number of ρ−connectable vertices from a fixed vertex, the second is whether the 1-p ercolation critical probability is equal to the Bernoulli bond percolation critical probability, and finally the third is whether the probability of the existence of ρ−passable path of length exceeding n starting from a fixed vertex always decays exponentially in the subcritical phase.