academician of Academy of Sciences of Republic of Uzbekistan
Professor
Doctor of physico-mathematical sciences (2001)
Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
20.05.1970
Phone: +998 (71) 207 91 39
E-mail: ,
Keywords: dynamical systems,
trajectory theory of dinamical systems,
nonlinear operators and processes,
random walks in random environment,
$p$-adic analysis,
the Gibbs measures,
lattice models of statistical mechanics.quadratic stochastic operators,
simplex,
trajectory,
Volterra and non-Volterra stochastic operators.
UDC: 512.544.23, 517.2, 517.219, 517.958, 517.98, 517.988.52, 517.998, 519.2, 519.21, 519.248, 530.1
MSC: 28d05, 28d15, 22d20, 22d25, 46l35, 60g99, 60b15, 82b20,92b05,97b10
Subject:
The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. For any normal subgroup of finite index the distribution of elements of the partition into conjugate classes on the Cayley tree is described. The notion of wood of Cayley trees is introduced and its group representation is discribed. The uniqueness of the translation-invariant Gibbs measure for the antiferromagnetic Potts model with an external field is proved. For $\lambda $-models the existence of three of translation-invariant and uncountable number of nontranslation-invariant Gibbs measurs on Cayley tree are proved. For any $\lambda$ formula of the critical temperature $ T_c (\lambda )$ is found. The existence of two translation invariant and uncountable numbers of the nontranslationally invariant extreme Gibbs measures for the Ising model with the compete interactions is roved and their constructive description is given. For Ising model with several spin values, Potts model and inhomogeneous Ising model sufficicy conditions of extremity of the unordered phase are obtained. For inhomogeneous Ising model it is proved that there exist only three periodic Gibbs measures corresponding to any normal subgroup of finite index and uncountable number of nonperiodic Gibbs measures. For $\lambda$-models it is proved that there exist only periodic Gibbs measures with period 2 corresponding to any normal subgroups of finite index. On wood of Cayley trees for inhomogeneous Ising model the periodic Gibbs measures are constructed. Using these measures for inhomegeneous Ising model on Cayley tree a new class of Gibbs measures is constructed. A random walk in a random environment on a class of metric spaces are defined. For random walks in a periodic random environment in case when for unit time the particle can transpose to finite distance and for random walks in any random environment when for unit time the particle can transpose only to the neighbouring points, sufficient conditions of non reflexivity are discribed. For example, we consider $Z^d$, infinite trees, free groups.
Main publications:
1. Rozikov U.A. Gibbs measures in biology and physics:
The Potts model. World Sci. Publ. Singapore. 2022, 368 pp.
2. Rozikov U.A. Population dynamics: algebraic and probabilistic approach. World Sci. Publ. Singapore. 2020, 460 pp.
3. Rozikov U.A. An introduction to mathematical billiards. World Sci. Publ. Singapore. 2019, 224 pp.
4. Ibragimov Z., Levenberg N., Rozikov U., Sadullaev A. (Eds). Algebra, Complex Analysis, and Pluripotential Theory. Springer Proceedings in Mathematics and Statistics. 2018, V. 264, Springer Nature Switzerland.
5. Rozikov U.A. Gibbs measures on Cayley trees. World Sci. Publ. Singapore. 2013, 404 pp.
