The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability These results were proven in the 1970s An open question since that time is whether Kozlov’s theorem holds in the shift-invariant setting, equivalently whether Sullivan’s conclusion can be improved from variation-summability to norm-summability We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov’s theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints We also present an extended version of Sullivan’s theorem Aside from simplifying some of the arguments in the original proof, our new version of Sullivan’s theorem applies in various settings not covered by the original proof In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional q-coloring shift for 
 $$q\ge 6$$
 

Gibbsian Representations of Continuous Specifications: The Theorems of Kozlov and Sullivan Revisited

This chapter focuses on the phase diagrams for classical lattice systems Peierls's method of contours. The conditional Gibbs distribution does not depend on the boundary condition, and the individual spin variables φ, V, are statistically independent with common distribution χ. The limiting procedure is trivial, and the unique limit Gibbs distribution represents a family of independent and χ-distributed random variables. For Hamiltonians with a finite set of periodic ground states satisfying a Peierls's stability condition, it can be shown that at large values of β, there appear some indecomposable and periodic limit Gibbs distributions called pure phases, and typical configurations of any pure phase are obtainable by small local distortions of one of the ground states. The coexistence of two or more pure phases appears together with the degeneration of the ground state of the basic Hamiltonian has at least two ground states in the domain of phase transitions.

Phase diagrams for classical lattice systems peierls's method of contours

We show that the problem of prescribing a system of random variables by means of conditional distributions can be considered from the algebraic point of view as a problem of consistency of an appropriate infinite system of linear equations. We demonstrate also that a potential energy (transition energy field) and probability (specification) are connected as the solutions of corresponding adjoint infinite systems of linear equations.

On a Class of Infinite Systems of Linear Equations Originating in Statistical Physics

The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov's theorem holds in the shift-invariant setting, equivalently whether Sullivan's conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov's theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan's theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan's theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional $q$-coloring shift for $q\geq 6$.

Gibbsian representations of continuous specifications: the theorems of Kozlov and Sullivan revisited

In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness of a finite random field with a given system of one-point conditional distributions. Using the axiomatic (without the notion of potential) definition of Hamiltonian, we show that any finite random field is Gibbsian. We also apply the proposed approach to Markov random fields. 

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On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form

In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field and one-point transition energy field. The former of them, namely the transition energy function, is a generalization of the notion of relative Hamiltonian introduced by Pirogov and Sinai. However, unlike the (relative) Hamiltonian, our objects are defined axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical definition of the Hamiltonian without involving the notion of potential, to propose a justification of the Gibbs formula for infinite systems and to answer the problem stated by D. Ruelle of how wide the class of specifications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics.

S Dachian

Papers

On the relationship of energy and probability in models of classical statistical physics