Theoretical notes on dense polymers in two dimensions.

Two models of dense two-dimensional (2d) polymers are considered: (1) when chain intersections in 2d are totally forbidden, and (2) when they are allowed to some extent. It is shown that both polymer chain statistics and dynamics are entirely different for the two models. In the first case studied by Duplantier in 1986 polymer chains are essentially segregated and are characterized by non-classical gamma exponents. The contact line between segregated chains is fractal which leads to an unusual demixing behavior in 2d blends. In the second case (crossings are allowed) polymer coils are overlapping and show mean-field statistics with logarithmic corrections. The correlation function of concentration fluctuations in this system is predicted to exhibit a universal long range power tail (1/r4) which is due to non-mean-field effects. The dynamical behavior of the two models is even more drastically different: The first model is characterized by a relatively fast dynamics with conformational relaxation time tN is proportional to N(15/8) (i.e. tN is slightly shorter than the Rouse time is proportional to N2). On the other hand an exponentially slow dynamics is predicted for model 2 (with 3d entanglements).