Chromatic Zeros on the Limit $G_{\infty }^{(p,\ell )}$ of the Family $G_m^{(p,\ell )}$ of Hierarchical Graphs

Abstract

We calculate the continuous accumulation set ${\mathcal{B}}_q(p,\ell )$ of zeros of the chromatic polynomial $P(G_m^{(p,\ell )},q)$ in the limit $m\to \infty$, on a family of graphs $G_m^{(p,\ell )}$ defined such that $G_m^{(p,\ell )}$ is obtained from $G_{m-1}^{(p,\ell )}$ by replacing each edge (i.e., bond) on $G_m^{(p,\ell )}$ by $p$ paths each of length $\ell$ edges, starting with the tree graph $T_2$. Our method uses the property that the chromatic polynomial $P(G,q)$ of a graph $G$ is equal to the $v=-1$ evaluation of the partition function of the $q$-state Potts model, together with (i) the property that $Z(G_m^{(p,\ell )},q,v)$ can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of $Z(G_{m-1}^{(p,\ell )},q,v^{\prime })$, where $v^{\prime }=F_{(p,\ell ),q}(v)$ is a rational function of $v$ and $q$ and (ii) ${\mathcal{B}}_q(p,\ell )(v)$ is the locus in the complex $q$-plane that separates regions of different asymptotic behavior of the $m$-fold iterated RG transformation $F_{(p,\ell ),q}(v)$ in the $m\to \infty$ limit. Thus, our results involve calculations of region diagrams in the complex $q$-plane showing the type of behavior that occurs in the $m\to \infty$ limit of the $m$-fold iterated RG transformation mapping $F_{(p,\ell ),q}(v)$ starting with the initial value $v=v_0=-1$. Calculations are presented of the maximal point $q_c(G_{\infty }^{(p,\ell )})$ at which the locus ${\mathcal{B}}_q$ crosses the real-$q$ axis, as well as several other points at which, depending on $p$ and $\ell$, the locus ${\mathcal{B}}_q$ crosses this axis. We give explicit results for a variety of $(p,\ell )$ cases and observe a number of interesting features. Calculations of the ground-state degeneracy of the Potts antiferromagnet at $q_c(G_{\infty }^{(p,\ell )})$ are presented. This work extends a previous study with R. Roeder of the $(p,\ell )=(2,2)$ case to higher $p$ and $\ell$ values.

80 pages, 40 figures

References in corpus (2)

• Zeros of the Potts Model Partition Function on Sierpinski Graphs
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs