AN EXACTLY SOLUBLE MODEL OF DIRECTED POLYMERS WITH MULTIPLE PHASE-TRANSITIONS

Polymer chains with hard-core interaction on a two-dimensional lattice are modeled by directed random walks. Two models, one with intersecti ng walks (IW) and another with non-intersecting walks (NIW) are presen ted, solved and compared. The exact solution of the two models, based on a representation using Grassmann variables, leads, surprisingly, to the same analytic expression for the polymer density and identical ph ase diagrams. There are three different phases as a function of hoppin g probability and single site monomer occupancy, with a transition fro m the dense polymer system to a polymer liquid (A) and a transition fr om the liquid to an empty lattice (B). Within the liquid phase there e xists a self-dual line with peculiar properties. The derivative of pol ymer density with respect to the single site monomer occupancy diverge s at transitions A and B, but is smooth across and along the self-dual line. The density-density correlation function along the direction x, perpendicular to the axis of directedness has a power law decay 1/x2 in the entire liquid phase, in both models. The difference between the two models shows up only in the behavior of the correlation function along the self-dual line: it decays exponentially in the IW model and as 1/x4 in the NIW model.