Bethe ansatz for the three-layer Zamolodchikov model

This paper is a continuation of our previous work (solv-int/9903001). We obtain two more functional relations for the eigenvalues of the transfer matrices for the $sl(3)$ chiral Potts model at $q^2=-1$. This model, up to a modification of boundary conditions, is equivalent to the three-layer three-dimensional Zamolodchikov model. From these relations we derive the Bethe ansatz equations.Comment: 22 pages, LaTeX, 5 figure

Maximin-optimal sustainable growth with nonrenewable resource and externalities

I offer an approach linking a welfare criterion to the “sustainable development opportunities” of the economy. This implies a dependence of a criterion on the information about the current state. I consider the problem for the Dasgupta-Heal-Solow-Stiglitz model with externalities. The economy-linked criterion is constructed on an example of the maximin principle applied to a hybrid level-growth measure. This measure includes as special cases the conventional measures of consumption level and percent change as a measure of growth. The hybrid measure or geometrically weighted percent can be used for measuring sustainable growth as an alternative to percent. The closed form solutions are obtained for the optimal paths including the paths, dynamically consistent with the updates in reserve estimates.Essential nonrenewable resource, modified Hotelling Rule, economy-linked criterion, geometrically weighted percent, normative resource peak.

A constant-utility criterion linked to an imperfect economy affected by irreversible global warming

The question of formulation of a social planner criterion for an imperfect economy is examined using an example of a polluting economy negatively affected by growing temperature. Imperfection of the economy is expressed here in deviations from the optimal initial state. It is shown that a criterion not linked to a specific initial state almost always implies either unsustainable or inefficient paths in the economy. In this paper, I link the constant-utility criterion to the initial amount of the resource reserve. This criterion implies efficient resource use and the paths of utility asymptotically approaching some constants, which depend on the parameters of the temperature function. The criterion can be formulated for the cases when the reserve estimate changes over time and when the high level of temperature can cause extinction.Essential nonrenewable resource, imperfect polluting economy, economy-linked criterion, semisustainable development, semiefficient extraction.

The vertex formulation of the Bazhanov-Baxter Model

In this paper we formulate an integrable model on the simple cubic lattice. The $N$ -- valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov -- Baxter Model. In the case when $N=2$ we reproduce the Korepanov's and Hietarinta's solutions of the Tetrahedron equation as some special cases.Comment: 20 pages, LaTeX fil

Eight-vertex model and non-stationary Lame equation

We study the ground state eigenvalues of Baxter's Q-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the $\Delta=-1/2$ six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic P-function where the modular parameter $\tau$ plays the role of (imaginary) time. In the scaling limit the equation transforms into a ``non-stationary Mathieu equation'' for the vacuum eigenvalues of the Q-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painleve III equation.Comment: 11 pages, LaTeX, minor misprints corrected, references adde

The eight-vertex model and Painleve VI

In this letter we establish a connection of Picard-type elliptic solutions of Painleve VI equation with the special solutions of the non-stationary Lame equation. The latter appeared in the study of the ground state properties of Baxter's solvable eight-vertex lattice model at a particular point, $\eta=\pi/3$, of the disordered regime.Comment: 9 pages, LaTeX, submitted to the special issue on Painleve VI, Journal of Physics

Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results

We study a special anisotropic XYZ-model on a periodic chain of an odd length and conjecture exact expressions for certain components of the ground state eigenvectors. The results are written in terms of tau-functions associated with Picard's elliptic solutions of the Painlev\'e VI equation. Connections with other problems related to the eight-vertex model are briefly discussed.Comment: 18 page

Functional relations and nested Bethe ansatz for sl(3) chiral Potts model at q^2=-1

We obtain the functional relations for the eigenvalues of the transfer matrix of the sl(3) chiral Potts model for q^2=-1. For the homogeneous model in both directions a solution of these functional relations can be written in terms of roots of Bethe ansatz-like equations. In addition, a direct nested Bethe ansatz has also been developed for this case.Comment: 20 pages, 6 figures, to appear in J. Phys. A: Math. and Ge