Yu. G. Kondratiev

One-particle subspaces in the stochasticXY model

We study the spectrum of the generatorHβ of the Glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1/β. We construct two so-called one-particle subspacesH± forHβ and describe the spectrum of the generator in these subspaces. As a consequence we find time asymptotics of the correlations for the Glauber dynamics.

EUCLIDEAN GIBBS STATES OF QUANTUM LATTICE SYSTEMS

An approach to the description of the Gibbs states of lattice models of interacting quantum anharmonic oscillators, based on integration in infinite dimensional spaces, is described in a systematic way. Its main feature is the representation of the local Gibbs states by means of certain probability measures (local Euclidean Gibbs measures). This makes it possible to employ the machinery of conditional probability distributions, known in classical statistical physics, and to define the Gibbs state of the whole system as a solution of the equilibrium (Dobrushin–Lanford–Ruelle) equation. With the help of this representation the Gibbs states are extended to a certain class of unbounded multiplication operators, which includes the order parameter and the fluctuation operators describing the long range ordering and the critical point respectively. It is shown that the local Gibbs states converge, when the mass of the particle tends to infinity, to the states of the corresponding classical model. A lattice approximation technique, which allows one to prove for the local Gibbs states analogs of known correlation inequalities, is developed. As a result, certain new inequalities are derived. By means of them, a number of statements describing physical properties of the model are proved. Among them are: the existence of the long-range order for low temperatures and large values of the particle mass; the suppression of the critical point behavior for small values of the mass and for all temperatures; the uniqueness of the Euclidean Gibbs states for all temperatures and for the values of the mass less than a certain threshold value, dependent on the temperature.

DIFFEOMORPHISM GROUPS AND CURRENT ALGEBRAS: CONFIGURATION SPACE ANALYSIS IN QUANTUM THEORY

The constuction of models of non-relativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space Γ of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on Γ, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on ℝd.

Non-equilibrium stochastic dynamics in continuum: The free case

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process

Wick calculus for regular generalized stochastic functionals

M

Small-Mass Behavior of Quantum Gibbs States for Lattice Models with Unbounded Spins

on a Riemannian manifold

Existence and A Priori Estimates for Euclidean Gibbs States

X

Uniqueness Problem for Quantum Lattice Systems with Compact Spins

. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in

Selection-mutation balance models with epistatic selection

X

Ground state euclidean measures for quantum lattice systems on compact manifolds

such that, with probability one, infinitely many particles will arrive at this set at some time