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F. H. Haydarov

Publications and source records attributed to F. H. Haydarov.

At least 19 recordsLinked to original sources
arXiv2026

Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures

We introduce a {\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a $q$-specification and the associated $q$-equilibrium measures. These objects arise naturally from a family of non-linear $q$-stochastic operators acting on the space of probability measures. A $q$-equilibrium measure is characterized as a fixed point of such operators, providing a non-linear analogue of the Gibbs equilibrium in the sense of DLR. We establish general conditions ensuring the existence and uniqueness of $q$-equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of $q$-specifications with an empty set of $q$-equilibrium measures. We characterize the set of $q$-equilibrium measures by studying the dynamical systems generated by a class of $q$-stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple $q$-equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the $q$-specification formalism extends the DLR theory from linear to non-linear settings and opens a new direction in the study of Gibbs measures and equilibrium states of physical systems.

math-ph↗
arXiv2025

Infinite dimensional analogues of nilpotent and solvable Lie algebras

We study infinite-dimensional analogues of nilpotent and solvable Lie algebras, focusing on the classes of pro-nilpotent, residually nilpotent, pro-solvable and residually solvable Lie algebras. We extend classical triangularization results (Engel's and Lie's theorems) to the pro-setting and establish existence results for the pro-nilpotent radical in pro-solvable algebras and in certain residually solvable algebras. We adapt finite-dimensional construction methods to produce residually solvable extensions with a given pro-nilpotent radical under natural finiteness conditions. By analyzing derivations and maximal tori of pro-nilpotent algebras, we extend the notion of rank and show that, for pro-nilpotent algebras of maximal rank, every derivation of a maximal residually solvable extension is inner. Finally, we describe standard constructions (tensor and direct sum products, central extensions) that preserve pro-nilpotency.

math.RA↗
arXiv2025

Coupled Ising-Potts Model: Rich Sets of Critical Temperatures and Translation-Invariant Gibbs Measures

We consider a coupled Ising-Potts model on Cayley trees of order $ k \geq 2 $. This model involves spin vectors $ (s, σ) $, and generalizes both the Ising and Potts models by incorporating interactions between two types of spins: $s = \pm 1$ and $σ= 1, \dots, q$. It is applicable to a wide range of systems, including multicomponent alloys, spin glasses, biological systems, networks, and social models. In this paper, we find some translation-invariant splitting Gibbs measures (TISGMs) and show, for $k\geq 2$, that at sufficiently low temperatures, the number of such measures is at least $2^{q}+1$. This is not an exact upper bound; for $k=2$ and $q=5$, we demonstrate that the number of TISGMs reaches the exact bound of 335, which is much larger than $2^5+1=33$. We prove, for $q=5$ that there are 12 critical temperatures at which the number of TISGMs changes, and we provide the exact number of TISGMs for each intermediate temperature. Additionally, we identify temperature regions where three TISGMs, close to the free measure, are either extreme or non-extreme among all Gibbs measures. We also show that the coupled Ising-Potts model exhibits properties absent in the individual Ising and Potts models. In particular, we observe the following new phenomena: 1. In both the Ising and Potts models, if a Gibbs measure exists at some temperature $T_0$, then it exists for all $T<T_0$. However, in the coupled Ising-Potts model, some TISGMs may only exist at intermediate temperatures (neither very low nor very high). 2. The 5-state Potts model has three critical temperatures and up to 31 TISGMs. We show that for $q=5$, the coupled Ising-Potts model has four times as many critical temperatures and approximately 11 times as many TISGMs. Thus, our model modifies the phase structure more rapidly and exhibits a significantly richer class of splitting Gibbs measures.

math.FA↗
arXiv2025

Non-Gibbsian Multivariate Ewens Probability Distributions on Regular Trees

Ewens' sampling formula (ESF) provides the probability distribution governing the number of distinct genetic types and their respective frequencies at a selectively neutral locus under the infinitely-many-alleles model of mutation. A natural and significant question arises: ``Is the Ewens probability distribution on regular trees Gibbsian?" In this paper, we demonstrate that Ewens probability distributions can be regarded as non-Gibbsian distributions on regular trees and derive a sufficient condition for the consistency condition. This study lays the groundwork for a new direction in the theory of non-Gibbsian probability distributions on trees.

math.PR↗
arXiv2023

Gradient Gibbs measures with periodic boundary laws of a generalized SOS model on a Cayley tree

We consider Gradient Gibbs measures corresponding to a periodic boundary law for a generalized SOS model with spin values from a countable set, on Cayley trees. On the Cayley tree, detailed information on Gradient Gibbs measures for models of SOS type are given in \cite{3, 16,8,11} and we continue the works for the generalized SOS model. Namely, in this paper, the problem of finding Gradient Gibbs measures that correspond to periodic boundary laws is reduced to a functional equation and by solving the equation all Gradient Gibbs measures with 4 periodic boundary laws are found.

math.PR↗
arXiv2023

Gibbs measures on spatial systems on vertices of Cayley trees

There are many research works devoted to Gibbs measure for models on Cayley trees. Among these works, there are some works in which the general results are identical, but the considered models are various. In this article, we present the construction of Gibbs measures in the language of measure theory and reply to the question ``When can we construct Gibbs specifications?" Also, we present a new condition (convenient for verification) which is equivalent to the consistency condition of kernels. The obtained results in the article are general, not for a particular model. On the contrary, these results hold for some considered models on Cayley trees.

math.PR↗
arXiv2023

On positive fixed points of operator of Hammerstein type with degenerate kernel and Gibbs Measures

From \cite{re} From \cite{re} it is known that ``translation-invariant Gibbs measures" of the model with an uncountable set of spin values can be described by positive fixed points of a nonlinear integral operator of Hammerstein type. In \cite{enh2015, MSSX} there are main results on positive fixed points of the operator of Hammerstein type with degenerate kernels, but it was not solved the existence of Gibbs measures corresponding to the founded fixed points for constructed kernels. This paper is an investigation of the papers \cite{enh2015} and \cite{MSSX}. In this paper we construct new degenerate kernels of the Hammerstein operator by taking into account problems in the theory of Gibbs measure, i.e. each positive fixed point of the operator gives translational-invariant Gibbs measure.

math.FA↗
arXiv2023

New class of Gibbs measures for two state Hard-Core model on a Cayley tree

In this paper, we consider a Hard-Core $(HC)$ model with two spin values on Cayley trees. The conception of alternative Gibbs measure is introduced and translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant. In addition, we study free energy of the model.

math.PR↗
arXiv2022

A HC model with countable set of spin values: uncountable set of Gibbs measures

We consider a hard core (HC) model with a countable set $\mathbb{Z}$ of spin values on the Cayley tree. This model is defined by a countable set of parameters $λ_{i}>0, i \in \mathbb{Z}\setminus\{0\}$. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model.

math.DS↗
arXiv2021

Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws

For SOS (solid-on-solid) model with external field and with spin values from the set of all integers, on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (a function defined on vertices of Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs which correspond to periodic boundary laws.

math-ph↗
arXiv2020

Gradient Gibbs measures for the SOS model with integer spin values on a Cayley tree

In the present paper we continue the investigation from [1] and consider the SOS (solid-on-solid) model on the Cayley tree of order $k \geq 2$. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.

math-ph↗
arXiv2017

Fixed points of Lyapunov integral operators and Gibbs measures

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for four competing interactions of models with uncountable (i.e. $[0,1]$) set of spin values on a Cayley tree. And we shall prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of fixed points.

math.FA↗
arXiv2017

Lyapunov operator $\mathcal L$ with degenerate kernel and Gibbs measures

In this paper we'll give a connection between four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two and Lyapunov integral equation. Also we'll study fixed points of Lyapunov operator with degenerate kernel which each fixed point of the operator is correspond to a {\it translation-invariant} Gibbs measure.

math.FA↗
arXiv2016

Four Competing interactions for models with uncountable set of spin values on a Cayley Tree

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two. We reduce the problem of describing the "splitting Gibbs measures" of the model to the analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of translation-invariant Gibbs measures.

math-ph↗
arXiv2016

Translation-invariant and periodic Gibbs measures for Potts model on a Cayley tree

In this paper is studied ferromagnetic three states Potts model on a Cayley tree of order three and we give explicit formulas for translation-invariant Gibbs measures. Furthermore, we show that under some conditions on the parameter of the antiferromagnetic Potts model with q-states with zero external field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic (non translation-invariant) Gibbs measures.

math-ph↗