Limiting behavior of a percolation cluster in a multilayered random environment with breakdown

We consider a sequence ($n=1,2,\cdots$) of integer Markov chains $\{\omega_{n,t}\}_{t\geqslant 1}$ with discrete time describing the percolation process in a band of width $n$ of a multilayered random medium in which a flow (breakdown) already exists, and random variable $\omega_{n,t}$ equals to the width of the percolation cluster at time $t$. For each value of $n$ and given random percolation mechanism the Markov chain $\{\omega_{n,t}\}_{t\geqslant 1}$ has a limit stationary distribution, given by the random variable $\omega_n$. In the case when the width $n$ of the layers of the medium under consideration tends to infinity, the limit distribution of the random variable $\omega_{n}\sqrt{b/n}$ ($b$ is some constant) is found, which is the Rayleigh distribution.