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Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I: General theory and square-lattice chromatic polynomial. (English) Zbl 1100.82509

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 J. L. Jacobsen and J. Salas, J. Stat. Phys. 104, No. 3-4, 701–723 (2001; Zbl 1100.82501).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C15 Coloring of graphs and hypergraphs

Citations:

Zbl 1100.82501

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