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

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.