# Virchenko Yu. P.

### Majorant estimates for the percolation threshold of a Bernoulli field on a square lattice

Tolmacheva Yu. A., Virchenko Yu. P.

Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1315–1326

We suggest a method for obtaining a monotonically decreasing sequence of upper bounds of percolation threshold of the Bernoulli random field on $Z^2$. On the basis of this sequence, we obtain a method of
constructing approximations with the guaranteed exactness estimate for a percolation probability. We compute the first term $c_2 = 0,74683$ of the considered sequence.

### Probability Space of Stochastic Fractals

Shpilinskaya O. L., Virchenko Yu. P.

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1467-1484

We develop a general method for the construction of a probability structure on the space F of random sets in ℝ. For this purpose, by using the introduced notion of *c*-system, we prove a theorem on the unique extension of a finite measure from a *c*-system to the minimal *c*-algebra. The obtained structure of measurability enables one to determine probability distributions of the *c*-algebra of random events sufficient, e.g., for the so-called fractal dimensionality of random realizations to be considered as a measurable functional on F.

### Correlation inequalities for statistical models of lattice gas

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 765–773

We consider models of statistical mechanics of the type of lattice gas with attractive interaction of general kind. We propose a method for obtaining inequalities that connect multipoint correlation functions of different order. This method allows one, on the one hand, to strengthen similar inequalities, which can be obtained within the framework of the FKG method, and on the other hand, to obtain new inequalities. We introduce the notion of duality for models of lattice gas. We show that if, under the transformation *p* ⇒ 1 - *p*, the correlation inequalities for a model with attraction turn into the corresponding inequalities that are also satisfied, then the correlation functions of the dual model also satisfy the latter inequalities.