"In this talk, we discuss the evolution of the concept of the Gibbs measure, starting from its definition in a finite volume to the definition of a Gibbs random field that does not use the notion of potential. We also touch upon the question of non-Gibbsian random fields and Dobrushin's program on their Gibbsianess (see, for example, [4]).

All known approaches to defining a Gibbs random field were based on the notion of the Hamiltonian as the sum of interaction potentials. From the physical point of view, such a definition is very natural since it allows one to construct models of statistical physics with given properties. Meanwhile, from the mathematical point of view, it is necessary to have a definition of the Gibbs random field in terms of its intrinsic properties without the notion of the potential. Such a definition was given in the works [2, 3, 5], which made it possible to present the theory of Gibbs random fields in a purely probabilistic manner. The presence of a probabilistic definition allows investigating general questions for Gibbs random fields such as uniqueness, mixing properties and validity of limit theorems.

Note that, along the way, three important problems were solved: Dobrushin's problem on the description of random fields by systems of consistent one-point conditional distributions (see [1]), Ruelle's problem on the Gibbsian representation of specification (see [3]) and the problem of the description of a finite random field by its local characteristics (see [5]).

The talk is based on the joint works with Boris S. Nahapetian (Institute of Mathematics, NAS RA).

References

[1] S. Dachian and B.S. Nahapetian, Description of specifications by means of probability distributions in small volumes under condition of very week positivity. J. Stat. Phys. 117, 2004, 281-300

[2] S. Dachian and B.S. Nahapetian, On Gibbsiannes of Random Fields. Markov Process. Relat. Fields 15, 2009, 81-104

[3] S. Dachian and B.S. Nahapetian, On the relationship of energy and probability in models of classical statistical physics. Markov Process. Relat. Fields 25, 2019, 649-681

[4] R.L. Dobrushin and S.B. Shlosman, Gibbsian description of “non-Gibbsian” fields. Uspekhi Mat. Nauk 52, 1997, 45-58

[5] L. Khachatryan and B.S. Nahapetian, On the characterization of a finite random field by conditional distribution and its Gibbs form. J. Theor. Probab., 2019, 19 pp."

**Linda Khachatryan**