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Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I. General theory and square-lattice chromatic polynomial
- J. Stat. Phys
, 2001
"... We study the chromatic polynomials ( = zero-temperature antiferromagnetic Potts-model partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer ..."
Abstract
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Cited by 42 (6 self)
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We study the chromatic polynomials ( = zero-temperature antiferromagnetic Potts-model partition functions) PG(q) for m × n rectangular subsets of the square lattice, with m ≤ 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 → ∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B2,B3,B4,B5 are limiting points of partition-function zeros as n → ∞ whenever the strip width m is ≥ 7 (periodic transverse b.c.) or ≥ 8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B10) cannot be a chromatic root of any graph. Key Words: Chromatic polynomial; chromatic root; antiferromagnetic Potts model; square lattice; transfer matrix; Fortuin-Kasteleyn representation; Temperley-Lieb algebra; Beraha–Kahane–Weiss theorem; Beraha numbers.
Character decomposition of Potts model partition functions
"... We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L×N, with boundary conditions that are free in the L-direction and periodic in the N-direction. The decomposition of the partition function in terms of the characters K1+2l (with l = 0, ..."
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Cited by 13 (8 self)
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We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L×N, with boundary conditions that are free in the L-direction and periodic in the N-direction. The decomposition of the partition function in terms of the characters K1+2l (with l = 0,1,...,L) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the K1+2l thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly j clusters are non-contractible with respect to the periodic lattice direction, and a The Q-state Potts model on a graph G = (V, E) is defined initially for Q integer by the partition function
6 Character decomposition of Potts model partition functions. I. Cyclic geometry
, 2006
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