Statistics of unentangled polymer loops and primitive paths

The concepts of reptation and the tube model have been successfully used to describe the dynamics of a system of entangled polymers. Attempts to apply this model have given rise to questions about the statistics of a polymer, represented by a lattice random walk, and its entanglement with an obstacle net. We have determined the number of ways such a walk can form unentangled closed loops of various types. If one reels in a general random walk from its ends, pulling out unentangled loop, one is left with the so-called primitive path, which is taken to represent the path of the tube. The probability that an N step random walk has a K step primitive path has been calculated. Asymptotic formulas for this probability are presented.