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

Abstract We consider a sequence ( n = 1, 2, ⋯) of finite Markov chains { ω n , t } t ≥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 ω 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 { ω n , t } t ≥1 has a limit stationary distribution, corresponding to the random variable ω 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 variables Ω n <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msqrt> <m:mi>b</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo>/</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msqrt> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $\begin{array}{} \displaystyle \sqrt{b/n} \end{array}$ ( b is some constant) is found, which is the Rayleigh distribution.