We consider statistical mechanics models for polymers. A polymer is a large particle of monomers binding together with chemical bonds. Sophisticated polymer chains, such as proteins or DNA, may have a size of 1,000,000 repeated monomers. These long chains drift in the media, i.e. water, oil or other chemical solvents. Polymers collect the impurity of the media along the way, and this is the Hamiltonian that we are going to discuss. 

The probabilistic setting is the following. A random walk is used to describe the behavior of the polymer, in the meanwhile, the impurity is described by a randomly distributed field in the space. In the classical probability theory, the end of the random walk path has a scale of order square root of N (diffusive). In the random media, polymers may have different scale which is greater/less than the square root of N. This is so-called the superdiffusive/subdiffusive phenomenon. When the temperature varys, the scale may go from diffusive to superdiffusive/subdiffusive. We call that the system undergoes a phase transition.