Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

Details Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

1996-12-31

arxiv.org/abs/cond-mat/9612249v2

Phys. Rev. E55, 5165 (1997)

Physics

Condensed Matter

Statistical Mechanics

33 pages, Latex, 5 postscript figures, published version; includes further comments on large-q series

Scientific paper

10.1103/PhysRevE.55.5165

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of $P(G,q)$ with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$ are presented and compared with series expansions and bounds.

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