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Zbl 1100.82509
Salas, Jesús; Sokal, Alan D.
Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I: General theory and square-lattice chromatic polynomial.
(English)
[J] J. Stat. Phys. 104, No. 3-4, 609-699 (2001). ISSN 0022-4715; ISSN 1572-9613/e

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) $P_G(q)$ for $m\times n$ rectangular subsets of the square lattice, with $m\leq 8$ (free or periodic transverse boundary conditions) and $n$ arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin-Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when $n\to\infty$, which arise from the crossing in modulus of dominant eigenvalues (Beraha-Kahane-Weiss theorem). We also provide evidence that the Beraha numbers $B_2,B_3,B_4,B_5$ are limiting points of partition-function zeros as $n\to\infty$ whenever the strip width $m$ is $\geq 7$ (periodic transverse b.c.) or $\geq 8$ (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps $B_{10}$) cannot be a chromatic root of any graph. For Part II see {\it J. L. Jacobsen} and {\it J. Salas}, J. Stat. Phys. 104, No. 3-4, 701--723 (2001; Zbl 1100.82501).
MSC 2000:
*82B20 Lattice systems
05C15 Chromatic theory of graphs and maps

Keywords: chromatic polynomial; chromatic root; antiferromagnetic Potts model: square lattice; transfer matrix; Fortuin-Kasteleyn representation; Beraha-Kahane-Weiss theorem; Beraha numbers

Citations: Zbl 1100.82501

Cited in: Zbl 1246.82025

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