Interactions and pressure functionals for disordered lattice systems

The purpose of this paper is to provide a theoretical framework for disordered spin systems on a lattice, similar to that of classical statistical mechanics in the sense of Ruelle [Ru]. We prove the existence of a continuous pressure functional on a large Banach space of random interactions (highly generalizing the classical one) and formulate an analog of the variational principle.     

On the ergodic properties of Glauber dynamics

We show that if there is an infinite volume Gibbs measure which satisfies a logarithmic Sobolev inequality with local coefficients of moderate growth, then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm.     

Uniqueness of Gibbs states for generalP(ϕ)2-weak coupling models by cluster expansion

We consider quantum fields with weak coupling in two space-time dimensions. We prove that the set of their ultraregular Gibbs states consists of only one point and this point is an extremal Gibbs state.     

On equivalence of spin and field pictures of lattice systems

We investigate the spin and field systems on a lattice connected by the Kac-Siegert transform. It is shown that the structures of corresponding theories are equivalent (in the sense of isomorphy of space of Gibbs states and order parameters). Using the idea of equivalence of spin and field pictures, we exhibit a class of lattice systems possessing infinitely uncountably many ground states. The systems of this type with infinite-range, slow-decaying interactions are expected to have a spin-glass phase transition.     

The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice

Using a method based on the application of hypercontractivity we prove the strong exponential decay to equilibrium for a stochastic dynamics of unbounded spin system on a lattice.     

The phase transition in the discrete Gaussian chain with 1/r 2 interaction energy

We exhibit a phase transition from a rough high-temperature phase to a rigid (localized) low-temperature phase in the discrete Gaussian chain with 1/r ² interaction energy. This transition is related to a localization transition in the ground state for a quantum mechanical particle in a one-dimensional periodic potential, coupled to quantum 1/f noise.This paper is dedicated to J. L. Lebowitz on the occasion of his 60th birthday     

Log-Sobolev inequalities for infinite one dimensional lattice systems

It is shown that a unique Gibbs measure of infinite spin system with short range interaction on one dimensional lattice satisfies log-Sobolev inequality.Supported by SFB 237     

The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition

Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.During the period of this research, both authors were partially supported by grants DAAL 03-86-K-0171 and DMS-8913328     

Spin glasses with long-range interaction at high temperatures

We study the Ising andN-vector spin glasses with exchange couplings J=(J _ij ;i, jZ ^d ), which are independent random variables with EJ_ij=0 andEJ ⁿ _ij ⁿ n!¦i–j¦ ^–nd , forn, some finite constant >0, and >1/2. For sufficiently small, we show that forE-a.a.J there is a weakly unique, extremal, infinite-volume Gibbs measure _J for which the expectation of a single (component of) spin vanishes and which has the cluster property inL ₂(E) with the same decay as interaction. This work is based on results and methods of Fröhlich and Zegarlinski.