  We consider several models whose motivation arises from statistical
mechanics. We begin by investigating some families of distributions of
translation invariant subgraphs of some Cayley graphs, and in particular
subgraphs of the square lattice. We then discuss some properties of the
Spin-Glass model in that lattice. We continue in describing some properties of
the Spin-Glass models in some other graphs.
  The last two parts of this work are devoted to the understanding of two
dynamical processes on graphs. The first one is the well known zero-temperature
Glauber dynamics on some families of graphs.
  The second dynamics, which we call the Loop Dynamics, is a natural
generalization of the zero-temperature Glauber dynamics, which appears to have
some interesting properties. We analyzed some of its properties for planar
lattices, though the exact same techniques are applied for larger families of
graphs as well.
