J M Maillard

Symmetry relations in exactly soluble models

Symmetry relations such as the star-triangle or the inverse relation are very useful in determining the partition function of two-dimensional exactly soluble models. A common construction of the three-dimensional equivalents of these symmetry relations is presented. They are used to derive, in a geometric way applying simultaneously to different kinds of spin models, the consequent global properties, i.e. the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function. The usefulness of the inverse relation is illustrated by an application to the three-dimensional Ising model.

A criterion for disorder solutions of spin models

A simple criterion is given which provides disorder solutions for spin models of the Ising or Potts type. New disorder solutions are thus obtained, in particular for the Potts model on a Kagome lattice and for the general anisotropic Ising model, on a three-dimensional cubic lattice. The validity of the disorder solution, when extended outside the physical domain, is also discussed.

Inversion relations and disorder solutions on Potts models

The inversion symmetry and the automorphy group generated by the latter, when combined with the spatial symmetries, are studied for the anisotropic Potts models on the triangular and checkerboard lattices. The exact expression of the partition function, which is known on some particular disorder subvarieties of the triangular model, is checked and a generalisation is proposed for the checkerboard lattice. The automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties.

A disorder solution for a cubic Ising model

The exact expression of the partition function of a three-dimensional cubic Ising model, with nearest-neighbour interactions, is given, when a certain relation between the coupling constants of the model is satisfied. This disorder solution is then compared with a partially resummed high-temperature expansion of the partition function of the model. The constraints on this expansion, which result from the existence of the disorder solution and from the inversion relation are discussed.

Inverse functional relations for lattice models

Inverse functional equations are studied for the partition functions and for the correlation functions of various models. The validity of the inverse relation for the partition function is justified with the help of approximations by quasi-one-dimensional models defined on strips of increasing size. The possibility of determining the partition function through the use of the inverse and other symmetry relations, coupled to analyticity hypotheses, is briefly investigated on the two-dimensional Ising model, with a field. The group generated by the inverse relation and the spatial symmetries of the model is studied in a three-dimensional case. Inverse relations are also exhibited, firstly for two-point and then for n-point correlation functions. They are first put into evidence by a geometric approach and then verified on a particular high-temperature expansion.

Algebraic invariants in soluble models

Exhibits, on the known solutions of the triangle relation for the exactly solved models, some simplifying methods of recovering their parametrisation in terms of algebraic varieties. The relation to the automorphy group, generated by the inverse and spatial symmetries of the model, is also analysed.

Integrable mappings from matrix transformations and their singularity properties

Various examples of bi-rational transformations having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics, as well as bi-rational transformations originating from spin edge models, are analysed using the singularity confinement method. This method provides results concerning the integrable (or not) character of these bi-rational transformations in complete agreement with the results obtained by visualization methods as well as methods based on the analysis of algebraic invariants.

Disorder solutions for Ising and Potts models with a field

The exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard Ising and Potts models with a magnetic field are obtained. Different specialisations and extensions of these exact results are examined.

Staggered lattice models

Some staggered models are studied along three different lines: exact solvability (Yang-Baxter equations); the existence of other exact solutions called disorder solutions; their group of symmetries. The analysis, based on the commutation of row to row transfer matrices of arbitrary size N, suggests that the exactly solvable models are either of non-staggered type or of free-fermion staggered type. In contrast, disorder solutions can easily be exhibited for staggered models: the example of the Ashkin-Teller model is analysed. Finally, the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models.

Rigorous bounds and the replica method for products of random matrices

The authors first develop a method to obtain rigorous bounds for the Lyapounov exponent of products of a random matrix. When applied to a class of 1D problems. including localisation, it reproduces the correct scaling behaviour at the band edge and gives very good approximations of the prefactors. They then study analytically the successive moments of the distribution law for the trace of the random matrix product within the whole energy band. The band centre anomaly is found to affect the whole statistics of the problem and the exact anomalous value of the Lyapounov exponent is recovered through the replica trick.