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

Abstract

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained from $G^{(p,\ell)}_{m-1}$ by replacing each edge (i.e., bond) on $G^{(p,\ell)}_m$ 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 $Z(G^{(p,\ell)}_m,q,v)$ can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of $Z(G^{(p,\ell)}_{m-1},q,v')$, where $v'=F_{(p,\ell),q}(v)$ is a rational function of $v$ and $q$ and (ii) ${\cal B}_q(p,\ell)(v)$ 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 $q_c(G^{(p,\ell)}_\infty)$ at which the locus ${\cal B}_q$ crosses the real-$q$ axis, as well as several other points at which, depending on $p$ and $\ell$, the locus ${\cal B}_q$ 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 $q_c(G^{(p,\ell)}_\infty)$ are presented. This work extends a previous study with R. Roeder of the $(p,\ell)=(2,2)$ case to higher $p$ and $\ell$ values.