Partition and generating function zeros in adsorbing self-avoiding walks

The Lee-Yang theory of adsorbing self-avoiding walks is presented. It is shown that Lee-Yang zeros of the generating function of this model asymptotically accumulate uniformly on a circle in the complex plane, and that Fisher zeros of the partition function distribute in the complex plane such that a positive fraction are located in annular regions centered at the origin. These results are examined in a numerical study of adsorbing self-avoiding walks in the square and cubic lattices. The numerical data are consistent with the rigorous results; for example, Lee-Yang zeros are found to accumulate on a circle in the complex plane and a positive fraction of partition function zeros appear to accumulate on a critical circle. The radial and angular distribution of partition function zeros are also examined and it is found to be consistent with the rigorous results on the Fisher zeros of interacting lattice clusters.