Salas, Jesús; Sokal, Alan D. Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I: General theory and square-lattice chromatic polynomial. (English) Zbl 1100.82509 J. Stat. Phys. 104, No. 3-4, 609-699 (2001). 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). Cited in 2 ReviewsCited in 30 Documents MSC: 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 05C15 Coloring of graphs and hypergraphs 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 Software:na20 PDFBibTeX XMLCite \textit{J. Salas} and \textit{A. D. Sokal}, J. Stat. Phys. 104, No. 3--4, 609--699 (2001; Zbl 1100.82509) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n. a(0) = 1, a(n) = Fibonacci(2*n). It has the property that a(n) = 1*a(n-1) + 2*a(n-2) + 3*a(n-3) + 4*a(n-4) + ... Decimal expansion of 2 + 2*cos(2*Pi/7). Irregular triangle read by rows: T(n, k) gives the coefficients of x^k of the minimal polynomials of the algebraic number over the rationals rho(n)^2, with rho(n) = 2*cos(Pi/n), for n >= 1.