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70
The SwendsenWang process does not always mix rapidly
 Proc. 29th ACM Symp. on Theory of Computing
, 1997
"... The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The l ..."
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Cited by 51 (4 self)
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The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the SwendsenWang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the SwendsenWang process requires exponential time (in the size of the system) to approach equilibrium.
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. I. General theory and squarelattice chromatic polynomial
 J. Stat. Phys
, 2001
"... We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel 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 ..."
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Cited by 42 (6 self)
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We study the chromatic polynomials ( = zerotemperature antiferromagnetic Pottsmodel 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 FortuinKasteleyn representation. In particular, we extract the limiting curves of partitionfunction 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 partitionfunction 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; FortuinKasteleyn representation; TemperleyLieb algebra; Beraha–Kahane–Weiss theorem; Beraha numbers.
Exact Potts model partition functions for strips of the triangular lattice
 J. Stat. Phys
"... We present exact calculations of the partition function of the qstate Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width Ly = 2 and arbitrarily great length. In the infinitelength limit the thermodynamic properties are discussed. The continuous ..."
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Cited by 20 (6 self)
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We present exact calculations of the partition function of the qstate Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width Ly = 2 and arbitrarily great length. In the infinitelength limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values. We also give exact calculations of the zerotemperature partition function (chromatic polynomial) and W(q), the exponent of the groundstate entropy, for the Potts antiferromagnet for honeycomb strips
Bethe Ansatz for the TemperleyLieb loop model with open boundaries
 Magic in the spectra of the XXZ quantum chain with boundaries at ∆ = 0 and ∆ = −1/2,” Nucl. Phys. B729, 387 (2005) [hepth/0505062
"... We diagonalise the Hamiltonian of the TemperleyLieb loop model with open boundaries using a coordinate Bethe Ansatz calculation. We find that in the groundstate sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has ..."
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Cited by 18 (2 self)
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We diagonalise the Hamiltonian of the TemperleyLieb loop model with open boundaries using a coordinate Bethe Ansatz calculation. We find that in the groundstate sector of the loop Hamiltonian, but not in other sectors, a certain constraint on the parameters has to be satisfied. This constraint has a natural interpretation in the TemperleyLieb algebra with boundary generators. The spectrum of the loop model contains that of the quantum spin1/2 XXZ chain with nondiagonal boundary conditions. We hence derive a recently conjectured solution of the complete spectrum of this XXZ chain. We furthermore point out a connection with recent results for the twoboundary sineGordon model. 1
Todorov, I.T.: Hecke algebraic properties of dynamical Rmatrices. Application to related matrix algebras
 Journ. Math. Phys
, 1999
"... Hecke algebraic properties of dynamical Rmatrices. Application to related quantum matrix algebras L.K. Hadjiivanov a 1 ..."
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Cited by 13 (11 self)
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Hecke algebraic properties of dynamical Rmatrices. Application to related quantum matrix algebras L.K. Hadjiivanov a 1
Promotion and cyclic sieving via webs
 J. Algebraic Combin
"... Abstract. We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon. 1. ..."
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Cited by 13 (0 self)
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Abstract. We show that Schützenberger’s promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon. 1.
A little statistical mechanics for the graph theorist
, 2008
"... In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalen ..."
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Cited by 11 (2 self)
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In this survey, we give a friendly introduction from a graph theory perspective to the qstate Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zerotemperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.
Transfer matrices and partitionfunction zeros for antiferromagnetic Potts models. IV. Chromatic polynomial with . . .
, 2004
"... We study the chromatic polynomial PG(q) for m × n square and triangularlattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zerotemperature limit of the partition function for the antiferromagnetic qstate Potts model defined on the lattice G. We show how ..."
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Cited by 11 (6 self)
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We study the chromatic polynomial PG(q) for m × n square and triangularlattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zerotemperature limit of the partition function for the antiferromagnetic qstate Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin–Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex qplane in the limit n → ∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.