Biased interacting self-avoiding walks on the four-simplex lattice

We have studied self-avoiding walks with both bias (stiffness or antistiffness) and step-step interaction (repulsive or attractive) on the four-simplex lattice, a deterministic fractal structure. Exact renormalization equations for partial partition sums are obtained by the standard method. We discuss the complete phase diagram in the three-dimensional space of fugacity per step, per bend, and per nearest-neighbor interaction. The phase transition surface between low- and high-density states is separated into first- and second-order parts by a continuous line of tricritical points. The group structure generated by the recursions allows us to find detailed information about the singular behavior of the walk density, in addition to critical and tricritical exponents. For stiff walks (large bias energy), we discuss the crossover to flexible (unbiased) behavior in the long-chain limit. The location of this crossover is found to depend upon whether chain parameters are critical or noncritical