Partition and generating function zeros in adsorbing self-avoiding walks

TL;DR

This paper investigates the distribution of zeros of the generating function and partition function for adsorbing self-avoiding walks, providing both theoretical insights and numerical evidence for their asymptotic behavior in the complex plane.

Contribution

It demonstrates that Lee-Yang zeros asymptotically form a circle and Fisher zeros distribute in annular regions, supported by numerical studies on lattice models.

Findings

Lee-Yang zeros accumulate on a circle in the complex plane

A positive fraction of Fisher zeros are located in annular regions

Numerical data support the theoretical distribution of zeros

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 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.