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Self-avoiding walks are studied on the 3-simplex fractal lattice as a model of linear polymer conformations in a dilute, non-homogeneous solution. A model is supplemented with bending energies and attractive-interaction energies between non-consecutively visited pairs of nearest-neighboring sites (contacts). It captures the main features of a semi-flexible polymer subjected to variable solvent conditions. Hierarchical structure of the fractal lattice enabled determination of the exact recurrence equations for the generating function, through which universal and local properties of the model were studied. Analysis of the recurrence equations showed that for all finite values of the considered energies and non-zero temperatures, polymer resides in an expanded phase. Critical exponents of the expanded phase are universal and the same as those for ordinary self-avoiding walks on the same lattice found earlier. As a measure of local properties, the mean number of contacts per mean number of steps as well as persistence length, are calculated as functions of Boltzmann weights associated with bending energies and attractive interactions between contacts. Both quantities are monotonic functions of stiffness weights for fixed interaction, and in the limit of infinite stiffness the number of contacts decreases to zero, while the persistence length increases unboundedly., Comment: 19pages,15 figures

We study the model of close-packed dimers on planar lattices belonging to the family of modified rectangular (MR) fractals, whose members are enumerated by an integer $p\geq 2$, as well as on the non-planar 4-simplex fractal lattice. By applying an exact recurrence enumeration method, we determine the asymptotic forms for numbers of dimer coverings, and numerically calculate entropies per dimer in the thermodynamic limit, for a sequence of MR lattices with $2\leq p\leq8$ and for 4-simplex fractal. We find that the entropy per dimer on MR fractals is increasing function of the scaling parameter $p$, and for every considered $p$ it is smaller than the entropy per dimer of the same model on $4$-simplex lattice. Obtained results are discussed and compared with the results obtained previously on some translationally invariant and fractal lattices., Comment: 15 pages, 9 figures, 2 tables