Linda Khachatryan's Publications (Full profile)

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. SharedIt link: https://rdcu.be/cZrcw, published online 10 November 2022.

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

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. https://rdcu.be/bQf9Y

В докладе показывается, что задача описания случайных полей в терминах подсистемы спецификации, состоящей только из одноточечных распределений вероятностей (проблема Добрушина), сводится к вопросу продолжения некоторого функционала, определенного на парах конфигураций, отличающихся только в одной точке, на все пространство пар конфигураций. Такое продолжение возможно при выполнении определенных условий симметрии функционала.

The estimations for the rate of convergence in the central limit theorem for homogenous martingale--difference random fields on the d-dimensional integer lattice are obtained.

For the purpose of studying infinite systems of statistical physics, Dobrushin in [1,2] and, independently, Lanford and Ruelle in~\cite{LR} introduced the fundamental notion of Gibbs random field and established its most important properties. The definition of Gibbs random field is directly related to the problem of prescribing a system of random variables by means of a consistent system of conditional distributions (specification). In [3,4] Dachian and Nahapetian showed that for a system of conditional distributions to be a specification it is necessary and sufficient that its elements have a Gibbs form. This result establishes a straightforward relationship between the probabilistic notion of (Gibbs) random field and the physical notion of energy (Hamiltonian). In the given consideration, we show that the problem of prescribing an infinite 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 that a potential energy (transition energy field) and probability (specification) are connected as the solutions of corresponding adjoint infinite systems of linear equations. This is joint work with B.S. Nahapetian. References [1] Dobrushin R.L., The description of random eld by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13, 1968, 197{224 [2] Dobrushin R.L., Gibbs random fields for lattice systems with pair-wise interaction. Funct. Anal. Appl. 2, 1968, 292{301 [3] Lanford O.E., Ruelle D., Observables at infinity and states with short-range correlations in statistical mechanics. Commun. Math. Phys. 13, 1969, 194{215 [4] Dachian S., Nahapetian B.S., On the relationship of energy and probability in models of classical statistical physics (submitted to the Journal of Markov Processes and Related Fields)

An infinite system of linear algebraic equations, concerned with the problem of description of Gibbs specifications, is considered. The conditions of existence of the system's solutions, as well as the condition of the uniqueness of the solution, are given. Moreover, some properties of the solution which are important from the statistical physics' point of view are studied.

Martingale method is quite widely applied in multidimensional problems. One of the main advantages of this method is the ability to prove a new type of limit theorems for dependent random variables. Particularly, this method was successfully applied in the theory of Gibbs random fields. From the point of view of application of the martingale method in the mathematical statistical physics it is especially important the presence of the possibility to study the behaviour of a system at the critical point. In our talk we will demonstrate the capabilities of the martingale method for studying some models of the mathematical statistical physics.