Florian Sobieczky
Zur Anzahl der Zusammenhangskomponenten von Urbildern station\"arer, ergodischer Felder
Preprint series: Mathematica Gottingensis
MSC:
60J10 Markov chains with discrete parameter
Abstract: The number of connected components of inverse images of random
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.
Keywords: percolation theory, random ralks in random scenery, spectral theory of markov chains