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

TL;DR

The paper investigates how chromatic zeros accumulate for the hierarchical graph family $G^{(p,\ell)}_m$ as $m\to\infty$, by exploiting an exact real-space RG transformation of the Potts model partition function with $v' = F_{(p,\ell),q}(v)$. The continuous locus ${\cal B}_q(p,\ell)$ in the complex $q$-plane, separating RG-dominated regions, is mapped and analyzed across parity classes of $p$ and $\ell$, yielding precise real-axis crossing points $q_c$, $q_L$, and, in certain cases, interior points $q_{\rm int}$ or cusp points $q_x$. The work extends prior results for $(p,\ell)=(2,2)$ to many higher $(p,\ell)$ pairs, computing crossing points, Mandelbrot-like substructures, and ground-state degeneracies $W_c(p,\ell)$, thereby enriching understanding of chromatic zeros on hierarchical lattices and their statistical-mechanical interpretations. These findings illuminate how graph structure and RG dynamics govern zero distributions, with potential implications for colorability thresholds and zero-temperature antiferromagnetic entropy on fractal lattices.

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.

Figures (40)

• Figure 1: $G^{(p,\ell)}_m$ graphs with $(p,\ell)=(2,3)$ and $m=0, \ 1, \ 2$.
• Figure 2: $G^{(p,\ell)}_m$ graphs with $(p,\ell)=(3,2)$ and $m=0, \ 1, \ 2$.
• Figure 3: Chromatic region diagram and locus ${\cal B}_q$ for $(p,\ell)=(2,2)$.
• Figure 4: Chromatic region diagram and locus ${\cal B}_q$ for $(p,\ell)=(2,2)$, showing detailed structure for the real interval $1.2 < q < 1.7$ and associated area of the complex $q$-plane with ${\rm Im}(q) > 0$. This depicts part of the infinite sequence $S_\infty$ of crossings of the locus ${\cal B}_q$ on the real-$q$ axis. In this and similar detailed figures below, the corresponding area with ${\rm Im}(q) < 0$ is just the complex-conjugate and hence is not shown.
• Figure 5: Chromatic region diagram and locus ${\cal B}_q$ for $(p,\ell)=(2,2)$, showing detailed structure for the real interval $1.2 < q < 1.3$ and associated area of the complex $q$-plane with ${\rm Im}(q) > 0$.