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.
Critical point and percolation probability in a long range site percolation model on Zd
