Trees at an interface

A lattice tree at an interface between two solvents of different quality is examined as a model of a branched polymer at an interface. Existence of th e free energy is shown, and the existence of critical lines in its phase di agram is proven. In particular there is a line of first order transitions s eparating a positive phase from a negative phase (the tree being predominan tly on either side of the interface in these phases), and a curve of locali zation delocalization transitions which separate the delocalized positive a nd negative phases from a phase where the tree is localized at the interfac e. This model is generalized to a branched copolymer which is examined in a certain average quenched ensemble. Existence of a thermodynamic limit is s hown for this model, and it is also shown that the model is self-averaging. Lastly, a model of branched polymers interacting with one another through a membrane is considered. The existence of a limiting free energy is shown, and it is demonstrated that if the interaction is strong enough, then the two branched polymers will absorb on one another.