Disorder regimes of directed polymers: Lattice case versus tree case

Relation between disorder regimes of directed lattice polymers and tree polymers have been scrutinized in this thesis. In the first part of this thesis we introduced the polymer in chemistry and explained our model associated with polymer and two basic well-known random walks.In chapter two we will focus on directed polymers in random environment model (a.k.a. directed lattice polymers). Impurities are randomly distributed in the environment where our model is defined. We want to model polymer chains that are affected by these impurities. By correlating an energy to each polymer structure, we select a polymer randomly. The asymptotic behavior of the directed polymer (as its length goes to infinity) which depends on the dimension of the ambient space and temperature will be analyzed. Temperature controls phase transitions of our model. At high temperatures it has been observed that the random disorder does not have a solid effect. However, when the temperature is below the critical value, polymers move to a phase where the random disorder affects at all. We struggle to improve our understanding of both phases on lattices and on trees. Results for a canonical multiplicative cascades analogous with tree polymers are established in the third part. We compare sharp results that are known for trees with the ones known for lattices and try to answer the open problems on lattices especially the situation at the critical temperature.