arXiv subjects
Publications and source records attributed to N. N. Ganikhodjaev.
The work is devoted to gradient Gibbs measures (GGMs) of a SOS model with countable set $\mathbb Z$ of spin values and having alternating magnetism on Cayley trees. This model is defined by a nearest-neighbor gradient interaction potential. Using Külske-Schriever argument, based on boundary law equations, we give several $q$-height-periodic translations invariant GGMs for $q=2,3,4$.
Motivating by the China's five element philosophy (CFEP) we construct a permuted Volterra quadratic stochastic operator acting on the four dimensional simplex. This operator (depending on 10 parameters) is considered as an evolution operator for CFEP. We study the discrete time dynamical system generated by this operator. Mainly our results related to a symmetric operator (depending on one parameter). We show that this operator has a unique fixed point, which is repeller. Moreover, in the case of non-zero parameter, it has two 5-periodic orbits. We divide the simplex to four subsets: the first set consists a single point (the fixed point); the second (resp. third) set is the set of initial points trajectories of which converge to the first (resp. second) 5-periodic orbit; the fourth subset is the set of initial points trajectories of which do not converge and their sets of limit points are infinite and lie on the boundary of the simplex. We give interpretations of our results to CFEP.
In this paper we find a sufficient condition under which the operator of bisexual population is contraction and show that this condition is not necessary.
The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very general and rather difficult. Intuitively clear that the Peierls argument does work for any symmetric model. But contours defined in Pirogov-Sinai theory do not work if one wants to use Peierls argument for more general symmetric models. We give a new definition of contour which allows relatively easier prove the main result of the Pirogov-Sinai theory for symmetric models. Namely, our contours allow us to apply the classical Peierls argument (with contour removal operation).
In the paper a one-dimensional model with nearest - neighbor interactions $I_n, n\in \Z$ and spin values $\pm 1$ is considered. It is known that under some conditions on parameters $I_n$ the phase transition occurs for the model. We define a notion of "phase separation" point between two phases. We prove that the expectation value of the point is zero and its the mean square fluctuation is bounded by a constant $C(β)$ which tends to 1/4 if $β\to\infty$. Here $β=\frac{1}{T}$, $ T>0$-temperature.
In this paper we consider quadratic stochastic operators designed on finite Abelian groups. It is proved that such operators have the property of regularity.