V. A. Zagrebnov

Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures “shifted” by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.

On the parallel dynamics for the little-hopfield model

We propose a method (algorithm) for calculation of the explicit formulas for evolution of the main and the residual overlaps. It allows us to confirm the Gardner-Derrida-Mottishaw second-step formula for the main overlap and to go beyond to the next steps. We discuss the dynamical status of the Amit-Gutfreund-Sompolinsky formula for the main overlap and some computersimulation results.

Exactly soluble model for structural phase transition with a Gaussian type anharmonicity

Abstract A new exactly solvable model for structural phase transition is proposed. The thermodynamic properties and the phase transition in this model are studied and compared with some previous results.

The Trotter-Kato product formula for Gibbs semigroups

The trace-norm convergence of the Trotter-Lie product formula has recently been proved for particular classes of Gibbs semigroups. In the present paper we prove it for the whole generality including generalization of the product formula proposed by Kato.

A generalized quasiaverage approach to the description of the limit states of then-vector Curie-Weiss ferromagnet

A set of all limit (Gibbs) states is constructed for the ferromagneticn-vector Curie-Weiss model by means of a generalized quasiaverage method.

An exactly solvable model for metal-insulator phase transition

A simple model of electrons interacting with photons which displays a metal-insulator phase transition in the case of s.c. and b.c.c. tight-binding bands is studied. A proof is given that the model is exactly solvable in the thermodynamic limit.

Perturbations of Gibbs semigroups

We present the analytic perturbation theory for Gibbs semigroups in the case when perturbations of generators are relatively bounded. Analyticity with respect to perturbation and semigroup parameters in the Tr-norm topology is proved and the corresponding domains are described.

Spectral properties of Kirkwood-Salsburg and Kirkwood-Ruelle operators

A method for solving Kirkwood-type equations in Banach spacesEξ(Λ) andEξS(Λ) is applied to derive spectral properties of Kirkwood-Salsburg and Kirkwood-Ruelle operators in these spaces. For stable interactions these operators are shown to have, besides the point spectrum, a residual one. We establish also that the residual spectrum may disappear if a superstable (singular) interaction between particles is switched on. In this case the bounded Kirkwood-Salsburg operator is proved to have a zero Fredholm radius.

A nonpolynomial generalization of exactly soluble models in statistical mechanics

Abstract A rigorous approach to a class of N-particle systems with an interaction term of the form −Nϕ(AN) is proposed where ϕ(·) is an arbitrary well-behaved function and AN is an intensive observable. General conditions are formulated under which the original Hamiltonian is proved to be thermodynamically equivalent to a linear in the intensive observable trial Hamiltonian. The method can be applied to various statistical mechanical models, thus generating different subclasses of exactly soluble models, provided that the linearized systems have this property.

One-dimensional random field Ising model and discrete stochastic mappings

Previous results relating the one-dimensional random field Ising model to a discrete stochastic mapping are generalized to a two-valued correlated random (Markovian) field and to the case of zero temperature. The fractal dimension of the support of the invariant measure is calculated in a simple approximation and its dependence on the physical parameters is discussed.