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We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

The hard square model in statistical mechanics has been investigated for the case when the activity z is −1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues x being zero, roots of unity, or solutions of x3 = 4cos2(πm/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.

The superintegrable chiral Potts model has many resemblances to the Ising model, so it is natural to look for algebraic properties similar to those found for the Ising model by Onsager, Kaufman and Yang. The spontaneous magnetization ℳr can be written in terms of a sum over the elements of a matrix Sr. The author conjectured the form of the elements, and this conjecture has been verified by Iorgov et al. The author also conjectured in 2008 that this sum could be expressed as a determinant, and has recently evaluated the determinant to obtain the known result for ℳr. Here we prove that the sum and the determinant are indeed identical expressions. Since the order parameters of the superintegrable chiral Potts model are also those of the more general solvable chiral Potts model, this completes the algebraic calculation of ℳr for the general model.

Does Response Rate Matter? Journal Editors Use of Survey Quality Measures in Manuscript Publication Decisions

The spontaneous magnetization of a two-dimensional lattice model can be expressed in terms of the partition function $W$ of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin. For the superintegrable case of the chiral Potts model with cylindrical boundary conditions, W can be expressed in terms of reduced hamiltonians H and a central spin operator S. We conjectured in a previous paper that W can be written as a determinant, similar to that of the Ising model. Here we generalize this conjecture to any Hamiltonians that satisfy a more general Onsager algebra, and give a conjecture for the elements of S., 18 pages, one figure

Nonexperimental and experimental studies have shown a lack of association between survey effort and nonresponse bias. This does not necessarily mean, however, that additional effort could not reduce nonresponse bias. Theories on nonresponse would suggest the use of different recruiting methods for additional survey effort in order to address nonresponse bias. This study looks at changes in survey estimates as a function of making additional calls under the same protocol and additional calls under a different protocol. Respondents who were interviewed as a result of more than five call attempts were not significantly different on any of the key survey variables than those interviewed with fewer than five calls. Those interviewed under a different survey protocol, however, were different on 5 of 12 measures. Additional interviews under both the same and different protocols contributed to the reduction of total nonresponse error. In sum, the use of multiple protocols for part of the survey effort increased the response rate, changed point estimates, and achieved lower total nonresponse error. Future work is needed on optimizing survey designs that implement multiple survey protocols.

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., Comment: 19 pages, one figure; Corrections made between 28 March 2008 and 28 April 2008: (1) 2.10: q to p; (2) 3.1: epsilon to 0 (not infinity); (3) 5.29: p to q; (4) p14: sub-head: p, q to q,p; (5) p15: sub-head: p, q to q,p; (6) 7.5 second theta to -theta ; (7) before 7.6: make more explicit definition of lambda_j. Several other typos fixed later

Background Hepatitis A vaccine coverage estimates needed for surveillance and vaccine policy decisions are not readily available for children older than 35 months or for adolescents. This article reports methodology developed for obtaining such estimates by telephone survey with and without provider record verification. Methods A random-digit-dial telephone survey with provider verification was conducted in Arizona and Oregon in 2004–2005 to obtain coverage estimates for children aged 2.5 to 15 years based on parental reports from telephone survey data alone, and from multiple logistic regressions using both telephone survey and provider data. Analysis was performed during 2006. Results Vaccination information was collected from parents of 1266 children, and provider verification from 488. Telephone survey and provider record–based hepatitis A vaccine coverage (one or more doses) was 60% and 65%, respectively, in Arizona, and 39% and 26%, respectively, in Oregon. Children who were younger, lived in metropolitan areas, or were Hispanic or nonwhite had significantly higher coverage; parents with immunization records provided more-accurate information. While a logistic model–based estimator developed using both parent and provider data performed slightly better than the estimator based on parent data alone, they differed mostly in the subgroups that had small sample sizes. Conclusions These are the first statewide provider-verified hepatitis A vaccine coverage estimates for children older than 35 months and indicate that telephone survey estimates as developed using this methodology could prove useful for immunization surveillance activities if interpreted cautiously.

It has been known for some time that the Boltzmann weights of the chiral Potts model can be parametrised in terms of hyperelliptic functions. but as yet no such parametrisation has been applied to the partition and correlation functions. Here we show that for N = 3 the function S(tp) that occurs in the recent calculation of the order parameters can he expressed quite simply in terms of such a parametrisation.

We derive the free energy of the chiral Potts model by the infinite lattice ``inversion relation'' method. This method is non-rigorous in that it always needs appropriate analyticity assumptions. Guided by previous calculations based on exact finite-lattice functional relations, we find that in addition to the usual assumption that the free energy be analytic and bounded in some principal domain of the rapidity parameter space that includes the physical regime, we also need a much less obvious symmetry. We can then obtain the free energy by Wiener-Hopf factorization in the complex planes of appropriate variables. Together with the inversion relation, this symmetry relates the values of the free energy in all neighbouring domains to those in the principal domain., Comment: 27 pages, 4 figures

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice “inversion relation” method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T pq . In terms of the t p ,t q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N−1)-dimensional lattice (for N>2). The function T pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.

We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the six-vertex model and XXZ chain, and for the eight-vertex model. In particular we discuss the “beyond the equator,” infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.

We consider the sum of dichromatic polynomials over non-separable rooted planar maps, an interesting special case of which is the enumeration of such maps. We present some known results and derive new ones. The general problem is equivalent to the q-state Potts model randomized over such maps: it remains an open question whether this model exhibits a phase transition or critical behaviour.

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.

We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with no two adjacent 1's. For this case we use the powerful corner transfer matrix method to numerically evaluate the partition function per site, density and some near-neighbour correlations to high accuracy. In particular for the square lattice we obtain the partition function per site to 43 decimal places., Comment: 16 pages, 2 built-in Latex figures, 4 tables

This is the third in a series of papers in which we set up and discuss the functional relations for the ``split rapidity line'' correlation function in the N - state chiral Potts model. The order parameters of the model can be obtained from this function. Here we consider the case N = 3 and write the equations explicitly in terms of the hyperelliptic functions parametrization. We also present four-term low-temperature series expansions, which we hope will cast light on the analyticity properties needed to solve the relations. The problem remains unsolved, but we hope that this will prove to be a step in the right direction., Comment: 17 pages

Following the method of Jimbo, Miwa, and others, we obtain functional relations for the order parameters of the chiral Potts model. We have not yet solved these relations. Here we discuss their properties and show how one should beware of spurious solutions.

Paul Fendley has recently found a "parafermionic" way to diagonalise a simple solvable hamiltonian associated with the chiral Potts model. Here we indicate how this method generalizes to the $\tau_2$ model with open boundaries and make some comments., Comment: 14 pages, 1 figure

We explicitly calculate the free energy Ψ of the general solvableN-state chiral Potts model in the scaling region, forT

We calculate the interfacial tension of theN-state chiral Potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions. Our result is valid for the general physical model (with positive Boltzmann weights) and at all subcritical temperatures. The interfacial tension has been calculated previously for the superintegrable chiral Potts model with skewed boundary conditions. UsingZ-invariance, Baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model. We show that this is indeed the case.

We use Kaufman's spinor method to calculate the bulk, surface and corner free energies $f_b, f_s, f_s', f_c$ of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For $f_b, f_s, f'_s$ our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic case. We note that the corner free energy $f_c$ depends only on the elliptic modulus $k$ that enters the working, and not on the argument $v$, which means that VJ's conjecture applies for the full anisotropic model. The only aspect of this paper that is new is the actual derivation of $f_c$, but by reporting all four free energies together we can see interesting structures linking them., 43 pages, 2 figures, paper amended to acknowledge previous work

We consider the six-vertex model with anti-periodic boundary conditions across a finite strip. The row-to-row transfer matrix is diagonalised by the `commuting transfer matrices' method. {}From the exact solution we obtain an independent derivation of the interfacial tension of the six-vertex model in the anti-ferroelectric phase. The nature of the corresponding integrable boundary condition on the $XXZ$ spin chain is also discussed., 18 pages, LaTeX with 1 PostScript figure

Partition functions for the three-state critical Potts model on finite square lattices and for a variety of boundary conditions are presented. The distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite-lattice result. The partition functions are then used to test the finite-size scaling predictions of conformal and modular invariance.

There is now a whole field in mathematical physics concerned with solvable models in statistical mechanics, field theory, and related areas. We indicate the influence that Onsager's solution of the planar Ising model has had, and continues to have, on this field.

In 2011 I reviewed the calculation by Onsager and Kaufman of the spontaneous magnetization of the square-lattice Ising model, which Onsager announced in 1949 but never published. I have recently been alerted to further original papers that bear on the subject. It is quite clear that the draft paper on which I relied was indeed written by Onsager, who was working on the problem with Kaufman, and that they had two derivations of the result., Comment: Follow-up to arXiv:1103.3347

We obtain the interfadal tension of the general solvable N-state chiral Potts model. It has exponent @ = (N + 2)/(2N) in bath the horizontal and vertical directions, in agreement with the scaling relation 2p = 2 -U. 1. Jlltr~uctiou

A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a quasi-periodic distribution of up to eight different coupling constants. The duality transformation on this tiling is considered and the self-dual couplings are determined. Furthermore, we analyze the subclass of exactly solvable models in detail parametrizing the coupling constants in terms of four rapidity parameters. For those, the self-dual couplings correspond to the critical points which, as expected, belong to the Onsager universality class., Comment: 25 pages, 6 figures

Lars Onsager announced in 1949 that he and Bruria Kaufman had proved a simple formula for the spontaneous magnetization of the square-lattice Ising model, but did not publish their derivation. It was three years later when C. N. Yang published a derivation in Physical Review. In 1971 Onsager gave some clues to his and Kaufman's method, and there are copies of their correspondence in 1950 now available on the Web and elsewhere. Here we review how the calculation appears to have developed, and add a copy of a draft paper, almost certainly by Onsager and Kaufman, that obtains the result., 32 pages, including copies of a draft paper almost certainly by Onsager and Kaufman Minor additions and corrections in second version

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 Albertiniet al.

We consider the star-triangle relation and the form of its solutions. We present some simple parametrizations of the weight functions of the three-state chiral Potts model. This model does not have the “difference property”: we discuss the resulting difficulties in attempting to use the corner transfer matrix method for this model.

The solvable generalized chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model with the specific two- and three-spin interactions on the body centered cubic lattice. Here we report the results of a calculation of the partition function per site for the infinite lattice.

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.

Lars Onsager and Bruria Kaufman calculated the partition function of the Ising model exactly in 1944 and 1949. Since then there have been many developments in the exact solution of similar, but usually more complicated, models. Here I shall mention a few, and show how some of the latest work seems to be returning once again to the properties observed by Onsager and Kaufman., 28 pages, 5 figures, section on six-vertex model revised

In this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models. The first is the Zamolodchikov model, which is the only nontrivial model on a three-dimen-sional lattice so far solved. The second is the chiral Potts model on the square lattice and its generalization associated with theU q(sl(n)) algebra, which is of current interest due to its connections with high-genus algebraic curves and with representations of quantum groups at roots of unity. We show that this last “sl(n)-generalized chiral Potts model” can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square-lattice layers with anN- valued (N⩾2) spin at each site. Further, in theN=2 case this three-dimen-sional model reduces (after a modification of the boundary conditions) to the Zamolodchikov model we mentioned above.

For the Ising model, the calculation of the spontaneous magnetization leads to the problem of evaluating a determinant. Yang did this by calculating the eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant and used Szego's theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D_PQ can be expressed as that of a product of two Cauchy-like matrices. One can then use the elementary exact formula for the Cauchy determinant. One of course regains the known result, originally conjectured in 1989., Comment: 16 pages, no figures; revised 11 Jan 2010 to correct citations and to include reference to subsequent work

We present some symmetry and factorization relations satisfied by the corner transfer matrices (CTMs) of the chiral Potts model. We show how the single-spin expectation values can be expressed in terms of the CTMs, and in terms of the related boost operator. Low-temperature calculations lead naturally to the variables that uniformize the Boltzmann weights of the model.

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically., 25 pages, one figure, last reference completed. Various typos fixed. Changes on 12 July 2008: Fig 1, 0 to +1; before (2.1), if to is; after (4.6), from to form; before (4.46), first three to middle two; before (4.46), last to others; Conclusions, 2nd para, insert how ; renewcommand \i to be \rm i

We consider the recently obtained functional relations for the transfer matrix of the solvable chiral Potts model. We show how these can be solved for the eigenvalues and hence obtain a more explicit result for the free energy than that already known. No uniformizing substitutions are used.

It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.

A two-dimensional quantum Hamiltonian ℋ N,M commuting with the layer-to-layer transfer matrix of the three-dimensional Zamolodchikov model is derived. This Hamiltonian is defined on a lattice ofN×M sites. The special casesN×2, 2×M, and 3×M are studied.

Hepatitis A vaccine coverage estimates needed for surveillance and vaccine policy decisions are not readily available for children older than 35 months or for adolescents. This article reports methodology developed for obtaining such estimates by telephone survey with and without provider record verification.A random-digit-dial telephone survey with provider verification was conducted in Arizona and Oregon in 2004-2005 to obtain coverage estimates for children aged 2.5 to 15 years based on parental reports from telephone survey data alone, and from multiple logistic regressions using both telephone survey and provider data. Analysis was performed during 2006.Vaccination information was collected from parents of 1266 children, and provider verification from 488. Telephone survey and provider record-based hepatitis A vaccine coverage (one or more doses) was 60% and 65%, respectively, in Arizona, and 39% and 26%, respectively, in Oregon. Children who were younger, lived in metropolitan areas, or were Hispanic or nonwhite had significantly higher coverage; parents with immunization records provided more-accurate information. While a logistic model-based estimator developed using both parent and provider data performed slightly better than the estimator based on parent data alone, they differed mostly in the subgroups that had small sample sizes.These are the first statewide provider-verified hepatitis A vaccine coverage estimates for children older than 35 months and indicate that telephone survey estimates as developed using this methodology could prove useful for immunization surveillance activities if interpreted cautiously.

Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models. They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model., Comment: 18 pages, 4 figures, for Proceedings of Conference on Symmetries and Integrability of Difference Equations. (SIDE VII), Melbourne, July 2006

The chiral Potts model continues to pose particular challenges in statistical mechanics: it is ``exactly solvable'' in the sense that it satisfies the Yang-Baxter relation, but actually obtaining the solution is not easy. Its free energy was calculated in 1988 and the order parameter was conjectured in full generality a year later. However, a derivation of that conjecture had to wait until 2005. Here we discuss that derivation., Comment: 22 pages, 3 figures, 29 references

We derive the order parameter of the chiral Potts model, using the method of Jimbo et al. The result agrees with previous conjectures., Comment: Version 2 submitted 21 Feb 2005. It has 7 pages, 2 figures. The introduction has been expanded and a significant typographical error in eqn 23 has been corrected

An outstanding problem in statistical mechanics is the order parameter of the chiral Potts model. An elegant conjecture for this was made in 1983. It has since been successfully tested against series expansions, but as far as the author is aware there is as yet no proof of the conjecture. Here we show that if one makes a certain analyticity assumption similar to that used to derive the free energy, then one can indeed verify the conjecture. The method is based on the ``broken rapidity line'' approach pioneered by Jimbo, Miwa and Nakayashiki., Comment: 29 pages, 7 figures. Citations made more explicit and some typos corrected

Elliott Lieb's ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the ``commuting transfer matrix'' $T, Q$ equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara-Miwa models., Comment: 30 pages, 6 figures

The normalization factor in the star-triangle relation can be evaluated in a simple form by taking determinants. If we combine this with the rotation symmetries, then we can show that a certain simple quantity I has to be independent of the rapidities. In this sense it is an invariant. We evaluate it for several particular models and find it is one for self-dual models, and is related to the modulus k (or k’) for the Ising, Kashiwara—Miwa and chiral Potts models.