  The main purpose of this thesis is to study the interplay between geometric
properties of infinite graphs and analytic and probabilistic objects such as
transition operators, harmonic functions and random walks on these graphs.
  For a transient random walk, there are several problems one is interested in:
for instance to study its convergence, to describe the bounded harmonic
functions for the random walk, to describe its Poisson boundary, or to study
the parameter of exponential decay of the transition probabilities of the
random walk.
  In the first part of the thesis we deal with similar problems in the context
of random walks on the so-called lamplighter graphs, which are wreath products
of graphs. The convergence and the Poisson boundary of lamplighter random walks
is studied for different underlying graphs, and the used methods are mostly of
a geometrical nature.
  In the second part of the thesis we consider Markov chains on directed,
labelled graphs. With such graphs we associate in a natural way a class of
infinite languages (sets of labels of paths in the graph) and we study the
growth sensitivity (or entropy sensitivity) of these languages using Markov
chains.
