**MSC:**- 60J10 Markov chains with discrete parameter
- 60D05 Geometric probability, stochastic geometry, random sets, See also {52A22, 53C65}

stationary, ergodic functions with discrete two-dimensional domain is

estimated, given only the functions' probability law: bounds for the

number of open clusters per vertex ($\kappa$) of inverse images of the

interval $[-\sigma, \sigma]$ for a threshhold $\sigma>0$ of a random function

on the set of nearest neighbours from $\Z^2$ are derived. Connectedness

refers to the ability to join two vertices from $\Z^2$ by an open path.

The existence of $\kappa$ in the stationary, ergodic case is proved. The

method used is that of a random walk on a randomly generated graph with

percolative structure. The estimates for $\kappa$ are tight in the number of steps

of the random walk. They involve the expected n-step return

probability, and the expected size of the number of bounding edges of the connected

component containing the origin. The spectral properties of the Markov process

allow these estimates by means of the Courant-Fischer min-max princple. The

shift of the unperturbed eigenvalues of a random walk on the homogeneous,

complete graph can be bounded, the pertubation is of finite rank, and

the rank equals the number of edges removed. An application is the i.i.d. case

of bond percolation.