The Experts below are selected from a list of 276 Experts worldwide ranked by ideXlab platform
Eric Vigoda - One of the best experts on this subject based on the ideXlab platform.
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swendsen wang algorithm on the mean field Potts Model
Random Structures and Algorithms, 2019Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) Θ(1) for β βc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0<βu<βrc that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the n-vertex complete graph satisfies: (i) Θ(1) for β<βu, (ii) Θ(n1/3) for β=βu, (iii) exp(nΩ(1)) for βu<β<βrc, and (iv) Θ(logn) for β≥βrc. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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ferromagnetic Potts Model refined bis hardness and related results
SIAM Journal on Computing, 2016Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for the hard-core Model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite $\Delta$-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree $\Delta$. To this end, we first present a detailed picture for the phase diagram for the infinite $\Delta$-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and order...
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swendsen wang algorithm on the mean field Potts Model
arXiv: Discrete Mathematics, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the $q$-state ferromagnetic Potts Model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case $q=2$, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) $\Theta(1)$ for $\beta \beta_c$, where $\beta_c$ is the critical temperature for the ordered/disordered phase transition. In contrast, for $q\geq 3$ there are two critical temperatures $0<\beta_u<\beta_{rc}$ that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the $n$-vertex complete graph satisfies: (i) $\Theta(1)$ for $\beta<\beta_u$, (ii) $\Theta(n^{1/3})$ for $\beta=\beta_u$, (iii) $\exp(n^{\Omega(1)})$ for $\beta_u<\beta<\beta_{rc}$, and (iv) $\Theta(\log{n})$ for $\beta\geq\beta_{rc}$. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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swendsen wang algorithm on the mean field Potts Model
International Workshop and International Workshop on Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising Model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0 =beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over.
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ferromagnetic Potts Model refined bis hardness and related results
arXiv: Computational Complexity, 2013Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts Model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic Models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts Model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.
R. J. Baxter - One of the best experts on this subject based on the ideXlab platform.
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A Conjecture for the Superintegrable Chiral Potts Model
Journal of Statistical Physics, 2008Co-Authors: R. J. BaxterAbstract:We adapt our previous results for the “partition function” of the superintegrable chiral Potts Model with open boundaries to obtain the corresponding matrix elements of e^− α H , where H is the associated Hamiltonian. The spontaneous magnetization ℳ_ r can be expressed in terms of particular matrix elements of e^− α H S _1 ^ r e^− β H , where S _1 is a diagonal matrix. We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts Model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising Model.
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a conjecture for the superintegrable chiral Potts Model
arXiv: Statistical Mechanics, 2008Co-Authors: R. J. BaxterAbstract:We adapt our previous results for the ``partition function'' of the superintegrable chiral Potts Model with open boundaries to obtain the corresponding matrix elements of e^{-\alpha H}, where H is the associated hamiltonian. The spontaneous magnetization M_r can be expressed in terms of particular matrix elements of e^{-\alpha H} S^r_1 \e^{-\beta H}, where S_1 is a diagonal matrix.We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts Model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising Model.
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derivation of the order parameter of the chiral Potts Model
Physical Review Letters, 2005Co-Authors: R. J. BaxterAbstract:We derive the order parameter of the chiral Potts Model, using the method of Jimbo et al. The result agrees with previous conjectures.
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Free energy of the chiral Potts Model in the scaling region
Journal of Statistical Physics, 1996Co-Authors: R. J. BaxterAbstract:We explicitly calculate the free energy Ψ of the general solvable N -state chiral Potts Model in the scaling region, for T
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Chiral Potts Model with skewed boundary conditions
Journal of Statistical Physics, 1993Co-Authors: R. J. BaxterAbstract:We obtain the transfer matrix functional relations for the chiral Potts Model with skewed boundary conditions and find that they are the same as for periodic boundary conditions, but with modified selection rules. As a start toward calculating the interfacial tension in general, we here evaluate it in a low-temperature limit, performing a Bethe-ansatz-type calculation. Finally, we specialize the relations to the superintegrable case, verifying the ansatz proposed by Albertini et al.
Andreas Galanis - One of the best experts on this subject based on the ideXlab platform.
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swendsen wang algorithm on the mean field Potts Model
Random Structures and Algorithms, 2019Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) Θ(1) for β βc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0<βu<βrc that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the n-vertex complete graph satisfies: (i) Θ(1) for β<βu, (ii) Θ(n1/3) for β=βu, (iii) exp(nΩ(1)) for βu<β<βrc, and (iv) Θ(logn) for β≥βrc. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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ferromagnetic Potts Model refined bis hardness and related results
SIAM Journal on Computing, 2016Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for the hard-core Model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite $\Delta$-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree $\Delta$. To this end, we first present a detailed picture for the phase diagram for the infinite $\Delta$-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and order...
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swendsen wang algorithm on the mean field Potts Model
arXiv: Discrete Mathematics, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the $q$-state ferromagnetic Potts Model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case $q=2$, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) $\Theta(1)$ for $\beta \beta_c$, where $\beta_c$ is the critical temperature for the ordered/disordered phase transition. In contrast, for $q\geq 3$ there are two critical temperatures $0<\beta_u<\beta_{rc}$ that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the $n$-vertex complete graph satisfies: (i) $\Theta(1)$ for $\beta<\beta_u$, (ii) $\Theta(n^{1/3})$ for $\beta=\beta_u$, (iii) $\exp(n^{\Omega(1)})$ for $\beta_u<\beta<\beta_{rc}$, and (iv) $\Theta(\log{n})$ for $\beta\geq\beta_{rc}$. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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swendsen wang algorithm on the mean field Potts Model
International Workshop and International Workshop on Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising Model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0 =beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over.
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ferromagnetic Potts Model refined bis hardness and related results
arXiv: Computational Complexity, 2013Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts Model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic Models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts Model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.
Daniel Stefankovic - One of the best experts on this subject based on the ideXlab platform.
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swendsen wang algorithm on the mean field Potts Model
Random Structures and Algorithms, 2019Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) Θ(1) for β βc, where βc is the critical temperature for the ordered/disordered phase transition. In contrast, for q≥3 there are two critical temperatures 0<βu<βrc that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the n-vertex complete graph satisfies: (i) Θ(1) for β<βu, (ii) Θ(n1/3) for β=βu, (iii) exp(nΩ(1)) for βu<β<βrc, and (iv) Θ(logn) for β≥βrc. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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ferromagnetic Potts Model refined bis hardness and related results
SIAM Journal on Computing, 2016Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for the hard-core Model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite $\Delta$-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree $\Delta$. To this end, we first present a detailed picture for the phase diagram for the infinite $\Delta$-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and order...
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swendsen wang algorithm on the mean field Potts Model
arXiv: Discrete Mathematics, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the $q$-state ferromagnetic Potts Model on the $n$-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. Long et al. studied the case $q=2$, the Swendsen-Wang algorithm for the mean-field ferromagnetic Ising Model, and showed that the mixing time satisfies: (i) $\Theta(1)$ for $\beta \beta_c$, where $\beta_c$ is the critical temperature for the ordered/disordered phase transition. In contrast, for $q\geq 3$ there are two critical temperatures $0<\beta_u<\beta_{rc}$ that are relevant. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts Model on the $n$-vertex complete graph satisfies: (i) $\Theta(1)$ for $\beta<\beta_u$, (ii) $\Theta(n^{1/3})$ for $\beta=\beta_u$, (iii) $\exp(n^{\Omega(1)})$ for $\beta_u<\beta<\beta_{rc}$, and (iv) $\Theta(\log{n})$ for $\beta\geq\beta_{rc}$. These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model.
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swendsen wang algorithm on the mean field Potts Model
International Workshop and International Workshop on Approximation Randomization and Combinatorial Optimization. Algorithms and Techniques, 2015Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric VigodaAbstract:We study the q-state ferromagnetic Potts Model on the n-vertex complete graph known as the mean-field (Curie-Weiss) Model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts Model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising Model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0 =beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts Model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over.
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ferromagnetic Potts Model refined bis hardness and related results
arXiv: Computational Complexity, 2013Co-Authors: Andreas Galanis, Daniel Stefankovic, Eric Vigoda, Linji YangAbstract:Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts Model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts Model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts Model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic Models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts Model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.
Jesper Lykke Jacobsen - One of the best experts on this subject based on the ideXlab platform.
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Integrable boundary conditions in the antiferromagnetic Potts Model
JHEP, 2020Co-Authors: Niall F. Robertson, Jesper Lykke Jacobsen, Michal Pawelkiewicz, Hubert SaleurAbstract:We present an exact mapping between the staggered six-vertex Model and an integrable Model constructed from the twisted affine $ {D}_2^2 $ Lie algebra. Using the known relations between the staggered six-vertex Model and the antiferromagnetic Potts Model, this mapping allows us to study the latter Model using tools from integrability. We show that there is a simple interpretation of one of the known K -matrices of the $ {D}_2^2 $ Model in terms of Temperley-Lieb algebra generators, and use this to present an integrable Hamiltonian that turns out to be in the same universality class as the antiferromagnetic Potts Model with free boundary conditions. The intriguing degeneracies in the spectrum observed in related works ([12, 13]) are discussed.
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Potts-Model critical manifolds revisited
Journal of Physics A: Mathematical and Theoretical, 2016Co-Authors: Christian R. Scullard, Jesper Lykke JacobsenAbstract:We compute critical polynomials for the q-state Potts Model on the Archimedean lattices, using a parallel implementation of the algorithm of Jacobsen (2014 J. Phys. A: Math. Theor 47 135001) that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size 6 × 6 unit cells, and the root in the temperature variable is determined numerically at q = 1 for bases of size 8 × 8. This leads to improved results for bond percolation thresholds, and for the Potts-Model critical manifolds in the real (q, v) plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts Model, and determine accurately the extent of the Berker–Kadanoff phase for the lattices studied.
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Potts-Model critical manifolds revisited
arXiv: Statistical Mechanics, 2015Co-Authors: Christian R. Scullard, Jesper Lykke JacobsenAbstract:We compute the critical polymials for the q-state Potts Model on all Archimedean lattices, using a parallel implementation of the algorithm of (Jacobsen, J. Phys. A: Math. Theor. 47 135001) that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size $6 \times 6$ unit cells, and the root in the temperature variable $v=e^K-1$ is determined numerically at $q=1$ for bases of size $8 \times 8$. This leads to improved results for bond percolation thresholds, and for the Potts-Model critical manifolds in the real $(q,v)$ plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the $(3,12^2)$ lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts Model, and determine accurately the extent of the Berker-Kadanoff phase for the lattices studied.
