A numerical study of the zeros of the grand partition function of $k$-mers on strips of width $k$

TL;DR

This study numerically analyzes the distribution of zeros of the grand partition function for $k$-mers on strips, revealing differences in critical behavior between dimers and trimers and highlighting the influence of geometry on statistical mechanics.

Contribution

It provides the first numerical investigation of zeros for $k$-mers on finite strips, showing distinct universality classes for trimers compared to dimers.

Findings

Zeros for $k=2$ match analytical predictions.

Zeros for $k=3$ are confined within a bounded region.

The structure of zeros indicates different critical behaviors.

Abstract

We study numerically, the distribution of the zeros of the grand partition function of $k$-mers on a $k \times L$ strip in the complex activity (z) plane. Using transfer matrix methods, we find that our results match the analytical predictions of Heilmann and Leib for $k = 2$. However, for $k = 3$, the zeros are confined within a bounded region, suggesting a fundamental difference in critical behavior. This indicates that trimers belong to a distinct universality class in some finite geometries. We observe that the density of zeros along multiple line segments in the complex plane reveals a richer structure than in the dimer case. {Our findings emphasize the role of geometric constraints in shaping the statistical mechanics of $k$-mer models and set the stage for further studies in higher-dimensional lattices.