Exact Solution of the Discrete (1+1)-dimensional RSOS Model in a Slit with Field and Wall Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with both walls. This model can also be mapped to a lattice polymer model of Motzkin paths in a slit interacting with both walls and including an osmotic pressure. This work generalises previous work on the RSOS model in the half-plane which has a solution that was shown recently to exhibit a novel mathematical structure involving basic hypergeometric functions ${}_3\phi_2$. Because of the mathematical relationship between half-plane and slit this work hence effectively explores the underlying $q$-orthogonal polynomial structure to that solution. It also generalises two other recent works: one on Dyck paths weighted with an osmotic pressure in a slit and another concerning Motzkin paths without an osmotic pressure term in a slit

Four-dimensional polymer collapse II: Interacting self-avoiding trails

We have simulated four-dimensional interacting self-avoiding trails (ISAT) on the hyper-cubic lattice with standard interactions at a wide range of temperatures up to length 4096 and at some temperatures up to length 16384. The results confirm the earlier prediction (using data from a non-standard model at a single temperature) of a collapse phase transition occurring at finite temperature. Moreover they are in accord with the phenomenological theory originally proposed by Lifshitz, Grosberg and Khokhlov in three dimensions and recently given new impetus by its use in the description of simulational results for four-dimensional interacting self-avoiding walks (ISAW). In fact, we argue that the available data is consistent with the conclusion that the collapse transitions of ISAT and ISAW lie in the same universality class, in contradiction with long-standing predictions. We deduce that there exists a pseudo-first order transition for ISAT in four dimensions at finite lengths while the thermodynamic limit is described by the standard polymer mean-field theory (giving a second-order transition), in contradiction to the prediction that the upper critical dimension for ISAT is $d_u=4$.Comment: 23 pages, 8 figure

The role of three-body interactions in two-dimensional polymer collapse

Various interacting lattice path models of polymer collapse in two dimensions demonstrate different critical behaviours. This difference has been without a clear explanation. The collapse transition has been variously seen to be in the Duplantier-Saleur $\theta$-point university class (specific heat cusp), the interacting trail class (specific heat divergence) or even first-order. Here we study via Monte Carlo simulation a generalisation of the Duplantier-Saleur model on the honeycomb lattice and also a generalisation of the so-called vertex-interacting self-avoiding walk model (configurations are actually restricted trails known as grooves) on the triangular lattice. Crucially for both models we have three and two body interactions explicitly and differentially weighted. We show that both models have similar phase diagrams when considered in these larger two-parameter spaces. They demonstrate regions for which the collapse transition is first-order for high three body interactions and regions where the collapse is in the Duplantier-Saleur $\theta$-point university class. We conjecture a higher order multiple critical point separating these two types of collapse.Comment: 17 pages, 20 figure

Monte Carlo Investigation of Lattice Models of Polymer Collapse in Five Dimensions

Monte Carlo simulations, using the PERM algorithm, of interacting self-avoiding walks (ISAW) and interacting self-avoiding trails (ISAT) in five dimensions are presented which locate the collapse phase transition in those models. It is argued that the appearance of a transition (at least) as strong as a pseudo-first-order transition occurs in both models. The values of various theoretically conjectured dimension-dependent exponents are shown to be consistent with the data obtained. Indeed the first-order nature of the transition is even stronger in five dimensions than four. The agreement with the theory is better for ISAW than ISAT and it cannot be ruled out that ISAT have a true first-order transition in dimension five. This latter difference would be intriguing if true. On the other hand, since simulations are more difficult for ISAT than ISAW at this transition in high dimensions, any discrepancy may well be due to the inability of the simulations to reach the true asymptotic regime.Comment: LaTeX file, 16 pages incl. 7 figure

Two-dimensional polymer networks at a mixed boundary: Surface and wedge exponents

We provide general formulae for the configurational exponents of an arbitrary polymer network connected to the surface of an arbitrary wedge of the two-dimensional plane, where the surface is allowed to assume a general mixture of boundary conditions on either side of the wedge. We report on a comprehensive study of a linear chain by exact enumeration, with various attachments of the walk's ends to the surface, in wedges of angles $\pi/2$ and $\pi$, with general mixed boundary conditions.Comment: 4 pages, Latex2e, 3 figures, Eur. Phys. J. B macro

Exact Solution of Semi-Flexible and Super-Flexible Interacting Partially Directed Walks

We provide the exact generating function for semi-flexible and super-flexible interacting partially directed walks and also analyse the solution in detail. We demonstrate that while fully flexible walks have a collapse transition that is second order and obeys tricritical scaling, once positive stiffness is introduced the collapse transition becomes first order. This confirms a recent conjecture based on numerical results. We note that the addition of an horizontal force in either case does not affect the order of the transition. In the opposite case where stiffness is discouraged by the energy potential introduced, which we denote the super-flexible case, the transition also changes, though more subtly, with the crossover exponent remaining unmoved from the neutral case but the entropic exponents changing