arXiv subjects
Publications and source records attributed to Yu. P. Virchenko.
The upper estimate of the percolation threshold of the Bernoulli random field on the hexagonal lattice is found. It is done on the basis of the cluster decomposition. Each term of the decomposition is estimated using the number estimate of cycles on the hexagonal lattice which represent external borders of possible finite clusters containing the fixed lattice vertex.
On the basis of the Kac formula of the characteristic function of the random variable defined by values of the quadratic functional $J_T [w]$ on trajectories of the standard Wiener process $\{w\}$, the method of its distribution density is proposed.
The Mandel probability distribution of one-mode stochastic radiation photocounts is analized. Approximations of n-photon registration probabilities with guaranteed accuracy are obtained in the case when the registration time is sufficiently small.
The more exact upper estimate of the percolation threshold for the {\it site problem} on the quadratic lattice ${\Bbb Z}^2$ have been found on the basis of the cluster decomposition. It is done by the number estimate of cycles on ${\Bbb Z}^2$ which maybe external boundaries of finite clusters
The problem of percolation along sites of square lattice is studied. The number of contours being external boundaries for finite clusters has been estimated using geometric considerations. This estimation makes it possible to determine more precisely the upper limit for the percolation threshold.