Adsorbing and collapsing trees

摘要

Self-interacting branched polymers interacting with a wall can be modelled by lattice trees in a half-space with a nearest-neighbour contact-fugacity beta and a visit-fugacity alpha conjugate to the number of visits the tree makes to the wall (which is the boundary of the half-space). We show that there is a limiting free energy in this model, and that it is a non-analytic function of the visit-fugacity for every finite and fixed value of the contact-fugacity. This implies the existence of an adsorption transition in this model, and there is a critical curve alpha(c)(+)(beta) in the phase diagram which separates the phase of desorbed trees from a phase of adsorbed trees. Moreover, we show that alpha(c)(+)(beta) > 0 for all finite values of the contact-fugacity, and the adsorption transition occurs at a strictly attractive value of the interaction between the tree and the wall. [References: 23]