Partition and generating function zeros in adsorbing self-avoiding walks

Abstract

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 centred 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 distributions of partition function zeros are also examined and it is found to be consistent with the rigorous results.