Recently, it has been shown that the zero-energy eigenstate – corresponding to the stationary state – of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can generically be written in the form of a so-called matrix-product state. We...

The symmetries of the partition function of a class of spin models with a transitive symmetry group on a bipartite graph or lattice are shown to be generated by a set of automorphisms of this permutation group and by the involutions of its center. The latter are generalized...

The multispin interactions for spin models with a transitive symmetry group and M states are considered. It is shown that the triangle-symmetric three-spin interactions depend only on the orbits in the set of all true triangles under the action of the symmetry group; this is analogous to the...

It is shown that the connective constant of a branching structure is modified if account is taken of the presence of a boundary containing a finite fraction of the vertices of the system. In particular, for a Cayley branch with branching ratio m, the calculation of the self-avoiding walk...

A necessary condition for the existence of a star-triangle transformation for a general, self-dual spin model is found. It is shown that this condition is, by symmetry, also sufficient for the Potts model and for the symmetric and general Ashkin-Teller models. For other models, explicit...

The concepts of duality and broken symmetry are discussed briefly. It is shown how generalized colouring polynomials can be used to calculate the terms in a high-temperature expansion effectively. These results and those from two previous papers are used to obtain high-temperature series for a...

Those self-avoiding polygons of the square lattice, which have a perimeter larger by 2D than the perimeter of their bounding rectangle and for which this is due to a “line defect”, are defined. Also, classes of “spiked” convex self-avoiding polygons are introduced. A large number of...

A transfer matrix technique is presented, which allows the explicit calculation of the generating functions for those self-avoiding polygons on the square lattice which have a perimeter equal to or two larger than the perimeter of their bounding rectangles.

All graphs and their contributions needed for a high-temperature expansion of the partition function of a completely permissible spin model up to order 16 for the square lattice and up to order 11 for the hypercubic lattice are listed. For the Potts model, the corresponding series are derived...

It is shown that the transitivity of the symmetry group of a classical spin model leads to a high-temperature expansion of the free energy of such a model in terms of multiply connected graphs. For models with a completely permissible symmetry group (e.g., for Ising and Potts models), these...