Boris Nahapetian's Publications (Full profile)

We develop a combinatorial approach to the description of lattice random fields with general (Polish) state space. This approach is based on so-called P-functions, which can be interpreted as a system of limiting correlation functions. We suggest a method of construction of P-functions, which can be applied in the theory of Gibbs random fields. Shared link: https://rdcu.be/czPBj

In this paper we present a new point of view on the mathematical foundations of statistical physics of innite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy eld and one-point transition energy eld. 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 dened axiomatically by their natural and physically well-founded intrinsic properties. The developed approach allowed us to give a proper mathematical denition of the Hamiltonian without involving the notion of potential, to propose a justication of the Gibbs formula for innite systems and to answer the problem stated by D. Ruelle of how wide the class of specications, which can be represented in Gibbsian form, is. Furthermore, this approach establishes a straightforward relationship between the probabilistic notion of (Gibbs) random eld and the physical notion of (transition) energy, and so opens the possibility to directly apply probabilistic methods to the mathematical problems of statistical physics.

We introduce sufficiently wide classes of non--linear functionals preserving normal (Gaussian) distribution and establish various conditions under which a sequence of such functionals is asymptotically normal. As one of a consequence, we obtain a generalization and sharpening of known results on the central limit theorem for weighted sums (linear functionals) of independent random variables.