We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with per... more We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.
We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 ... more We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 integer), in the complex-temperature plane. The models are defined on L × N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N ) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A p−1 type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N → ∞, of partition function zeros for p = 4, 5, 6, ∞ and L = 2, 3, 4 and study selected features for p > 6 and/or L > 4. This information enables us to formulate several conjectures about the thermodynamic limit, L → ∞, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p = 4). We show how this feature is modified by taking fixed transverse boundary conditions.
Childhood absence epilepsy (CAE) accounts for 10% to 12% of epilepsy in children under 16 years o... more Childhood absence epilepsy (CAE) accounts for 10% to 12% of epilepsy in children under 16 years of age. We screened for mutations in the GABA A receptor (GABAR) b3 subunit gene (GABRB3) in 48 probands and families with remitting CAE. We found that four out of 48 families (8%) had mutations in GABRB3. One heterozygous missense mutation (P11S) in exon 1a segregated with four CAE-affected persons in one multiplex, two-generation Mexican family. P11S was also found in a singleton from Mexico. Another heterozygous missense mutation (S15F) was present in a singleton from Honduras. An exon 2 heterozygous missense mutation (G32R) was present in two CAEaffected persons and two persons affected with EEG-recorded spike and/or sharp wave in a two-generation Honduran family. All mutations were absent in 630 controls. We studied functions and possible pathogenicity by expressing mutations in HeLa cells with the use of Western blots and an in vitro translation and translocation system. Expression levels did not differ from those of controls, but all mutations showed hyperglycosylation in the in vitro translation and translocation system with canine microsomes. Functional analysis of human GABA A receptors (a1b3-v2g2S, a1b3-v2[P11S]g2S, a1b3-v2[S15F]g2S, and a1b3-v2[G32R]g2S) transiently expressed in HEK293T cells with the use of rapid agonist application showed that each amino acid transversion in the b3-v2 subunit (P11S, S15F, and G32R) reduced GABA-evoked current density from whole cells. Mutated b3 subunit protein could thus cause absence seizures through a gain in glycosylation of mutated exon 1a and exon 2, affecting maturation and trafficking of GABAR from endoplasmic reticulum to cell surface and resulting in reduced GABA-evoked currents.
We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q an... more We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L t and for arbitrarily great length L ℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v)= $\sum {_{j = 1}^{N_{Z,G,\lambda } } c_{Z,G,j} (\lambda _{Z,G,j} )} ^{L_\ell}$ . We give general formulas for N Z, G, j and its specialization to v=−1 for arbitrary L t for both types of boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus $B$ , arising as the accumulation set of partition function zeros as L ℓ→∞, in the q plane for fixed v and in the v plane for fixed q.
We studied the peritoneal absorption and elimination of lntralipid after its intraperitoneal and ... more We studied the peritoneal absorption and elimination of lntralipid after its intraperitoneal and intravenous infusion into rats, by means of a two-compartment model. Follow-up measurements of the plasma triglyeeride rate were made.
We analyze the scaling and finite-size-scaling behavior of the two-dimensional 4-state Potts mode... more We analyze the scaling and finite-size-scaling behavior of the two-dimensional 4-state Potts model. We find new multiplicative logarithmic corrections for the susceptibility, in addition to the already known ones for the specific heat. We also find additive logarithmic corrections to scaling, some of which are universal. We have checked the theoretical predictions at criticality and off criticality by means of high-precision Monte Carlo data.
We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller mo... more We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (τint.δ≥ const xC H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio τint.δ/C H appears to tend to infinity either as a logarithm or as a small power (0.05≲p≲0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.
We study the 3-state square-lattice Potts antiferromagnet at zero temperature by a Monte Carlo si... more We study the 3-state square-lattice Potts antiferromagnet at zero temperature by a Monte Carlo simulation using the Wang-Swendsen-Kotecký cluster algorithm, on lattices up to 1024 × 1024. We confirm the critical exponents predicted by Burton and Henley based on the height representation of this model.
Using results from conformal field theory, we compute several universal amplitude ratios for the ... more Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ★=limL→∞ξ(L)/L and the first four magnetization moment ratios V 2n =〈 $M$ 2n 〉/〈 $M$ 2〉n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n . We confirm these predictions by a high-precision Monte Carlo simulation.
Using results from conformal field theory, we compute several universal amplitude ratios for the ... more Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ⋆ = lim L→∞ ξ(L)/L and the first four magnetization moment ratios V 2n = M 2n / M 2 n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G * 2n . We confirm these predictions by a high-precision Monte Carlo simulation.
We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhib... more We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq≥7), triangular lattice (q≥11), hexagonal lattice (q≥4), and Kagomé lattice (q≥6). The proofs are based on the Dobrushin uniqueness theorem.
We derive some new structural results for the transfer matrix of square-lattice Potts models with... more We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q −47 (resp. q −46). Finally, we compute chromatic roots for strips of widths 9≤m≤12 with free boundary conditions and locate roughly the limiting curves.
We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition fu... more We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P G (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 B 2,B 3,B 4,B 5 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 B 10) cannot be a chromatic root of any graph.
We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and ... more We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= $\sum _{j = 1}^{N_{Z,G,\lambda } }$ c z,G,j (λ z,G,j )m-1. We give general formulas for N Z,G,j and its specialization to v=−1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus ${\mathcal{B}}$ , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.
We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with per... more We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.
We study the chromatic polynomial P G (q) for m×n triangular-lattice strips of widths m≤12P,9F (w... more We study the chromatic polynomial P G (q) for m×n triangular-lattice strips of widths m≤12P,9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin–Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n→∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n→∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.
We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with per... more We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.
We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 ... more We study the phase diagram of Q-state Potts models, for Q = 4 cos 2 (π/p) a Beraha number (p > 2 integer), in the complex-temperature plane. The models are defined on L × N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N ) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A p−1 type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N → ∞, of partition function zeros for p = 4, 5, 6, ∞ and L = 2, 3, 4 and study selected features for p > 6 and/or L > 4. This information enables us to formulate several conjectures about the thermodynamic limit, L → ∞, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p = 4). We show how this feature is modified by taking fixed transverse boundary conditions.
Childhood absence epilepsy (CAE) accounts for 10% to 12% of epilepsy in children under 16 years o... more Childhood absence epilepsy (CAE) accounts for 10% to 12% of epilepsy in children under 16 years of age. We screened for mutations in the GABA A receptor (GABAR) b3 subunit gene (GABRB3) in 48 probands and families with remitting CAE. We found that four out of 48 families (8%) had mutations in GABRB3. One heterozygous missense mutation (P11S) in exon 1a segregated with four CAE-affected persons in one multiplex, two-generation Mexican family. P11S was also found in a singleton from Mexico. Another heterozygous missense mutation (S15F) was present in a singleton from Honduras. An exon 2 heterozygous missense mutation (G32R) was present in two CAEaffected persons and two persons affected with EEG-recorded spike and/or sharp wave in a two-generation Honduran family. All mutations were absent in 630 controls. We studied functions and possible pathogenicity by expressing mutations in HeLa cells with the use of Western blots and an in vitro translation and translocation system. Expression levels did not differ from those of controls, but all mutations showed hyperglycosylation in the in vitro translation and translocation system with canine microsomes. Functional analysis of human GABA A receptors (a1b3-v2g2S, a1b3-v2[P11S]g2S, a1b3-v2[S15F]g2S, and a1b3-v2[G32R]g2S) transiently expressed in HEK293T cells with the use of rapid agonist application showed that each amino acid transversion in the b3-v2 subunit (P11S, S15F, and G32R) reduced GABA-evoked current density from whole cells. Mutated b3 subunit protein could thus cause absence seizures through a gain in glycosylation of mutated exon 1a and exon 2, affecting maturation and trafficking of GABAR from endoplasmic reticulum to cell surface and resulting in reduced GABA-evoked currents.
We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q an... more We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L t and for arbitrarily great length L ℓ, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v)= $\sum {_{j = 1}^{N_{Z,G,\lambda } } c_{Z,G,j} (\lambda _{Z,G,j} )} ^{L_\ell}$ . We give general formulas for N Z, G, j and its specialization to v=−1 for arbitrary L t for both types of boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus $B$ , arising as the accumulation set of partition function zeros as L ℓ→∞, in the q plane for fixed v and in the v plane for fixed q.
We studied the peritoneal absorption and elimination of lntralipid after its intraperitoneal and ... more We studied the peritoneal absorption and elimination of lntralipid after its intraperitoneal and intravenous infusion into rats, by means of a two-compartment model. Follow-up measurements of the plasma triglyeeride rate were made.
We analyze the scaling and finite-size-scaling behavior of the two-dimensional 4-state Potts mode... more We analyze the scaling and finite-size-scaling behavior of the two-dimensional 4-state Potts model. We find new multiplicative logarithmic corrections for the susceptibility, in addition to the already known ones for the specific heat. We also find additive logarithmic corrections to scaling, some of which are universal. We have checked the theoretical predictions at criticality and off criticality by means of high-precision Monte Carlo data.
We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller mo... more We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (τint.δ≥ const xC H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio τint.δ/C H appears to tend to infinity either as a logarithm or as a small power (0.05≲p≲0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.
We study the 3-state square-lattice Potts antiferromagnet at zero temperature by a Monte Carlo si... more We study the 3-state square-lattice Potts antiferromagnet at zero temperature by a Monte Carlo simulation using the Wang-Swendsen-Kotecký cluster algorithm, on lattices up to 1024 × 1024. We confirm the critical exponents predicted by Burton and Henley based on the height representation of this model.
Using results from conformal field theory, we compute several universal amplitude ratios for the ... more Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ★=limL→∞ξ(L)/L and the first four magnetization moment ratios V 2n =〈 $M$ 2n 〉/〈 $M$ 2〉n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n . We confirm these predictions by a high-precision Monte Carlo simulation.
Using results from conformal field theory, we compute several universal amplitude ratios for the ... more Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ⋆ = lim L→∞ ξ(L)/L and the first four magnetization moment ratios V 2n = M 2n / M 2 n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G * 2n . We confirm these predictions by a high-precision Monte Carlo simulation.
We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhib... more We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq≥7), triangular lattice (q≥11), hexagonal lattice (q≥4), and Kagomé lattice (q≥6). The proofs are based on the Dobrushin uniqueness theorem.
We derive some new structural results for the transfer matrix of square-lattice Potts models with... more We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q −47 (resp. q −46). Finally, we compute chromatic roots for strips of widths 9≤m≤12 with free boundary conditions and locate roughly the limiting curves.
We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition fu... more We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P G (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 B 2,B 3,B 4,B 5 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 B 10) cannot be a chromatic root of any graph.
We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and ... more We present exact calculations of the Potts model partition function Z(G,q,v) for arbitrary q and temperature-like variable v on n-vertex strip graphs G of the triangular lattice for a variety of transverse widths equal to L vertices and for arbitrarily great length equal to m vertices, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form Z(G,q,v)= $\sum _{j = 1}^{N_{Z,G,\lambda } }$ c z,G,j (λ z,G,j )m-1. We give general formulas for N Z,G,j and its specialization to v=−1 for arbitrary L. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite triangular lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus ${\mathcal{B}}$ , arising as the accumulation set of partition function zeros as m→∞, in the q plane for fixed v and in the v plane for fixed q. Explicit results for partition functions are given in the text for L=3 (free) and L=3, 4 (cylindrical), and plots of partition function zeros and their asymptotic accumulation sets are given for L up to 5. A new estimate for the phase transition temperature of the q=3 Potts antiferromagnet on the 2D triangular lattice is given.
We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with per... more We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.
We study the chromatic polynomial P G (q) for m×n triangular-lattice strips of widths m≤12P,9F (w... more We study the chromatic polynomial P G (q) for m×n triangular-lattice strips of widths m≤12P,9F (with periodic or free transverse boundary conditions, respectively) and arbitrary lengths n (with free longitudinal boundary conditions). The chromatic polynomial gives the zero-temperature limit of the partition function for the q-state Potts antiferromagnet. We compute the transfer matrix for such strips in the Fortuin–Kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q-plane in the limit n→∞. We recompute the limiting curve obtained by Baxter in the thermodynamic limit m,n→∞ and find new interesting features with possible physical consequences. Finally, we analyze the isolated limiting points and their relation with the Beraha numbers.
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Papers by Jesus Salas