arXiv subjects

Mirmukhsin Makhmudov

Publications and source records attributed to Mirmukhsin Makhmudov.

7 recordsLinked to original sources

arXiv2026

On perturbations that preserve the connectivity properties in tree percolations

We consider a general bond percolation on an infinite locally finite tree, where the edge retention probabilities $p_e$ are replaced by $\min\{1,q_{|e|}p_e\}$, where $\{q_n\}_{n\ge 1}$ is a sequence of positive perturbation factors and $|e|$ denotes the distance between the edge $e$ and the root. If the original percolation model admits infinite clusters, it is of interest to investigate under which perturbations $0<q_n\le 1$ this connectivity property is preserved. Conversely, if the original percolation does not admit infinite clusters, we are led to study the stability of such a property under perturbations satisfying $q_n\ge 1$. In both cases, under minimal assumptions on the original model, we show that the percolative behaviour is stable against certain quantitative non-trivial perturbations. We also discuss an application of our results to the Erdős similarity conjecture for Cantor sets.

math.PR↗

arXiv2026

Effective Intrinsic Ergodicity for renewal-type potentials on S-gap shifts

We establish effective intrinsic ergodicity for renewal-type potentials on one-sided $S$-gap shifts. Inducing on the one-symbol cylinder $[1]$ reduces the system to a full shift over the alphabet $S$, where the induced potential becomes a one-symbol potential and the equilibrium measure is Bernoulli. The associated renewal equation has a unique solution $P$, and under the condition $P>ϕ(0^\infty)$ (automatic when $S$ is infinite), we show that $P$ is the topological pressure and that the potential admits a unique equilibrium state $μ_ϕ$. Our main result is an effective intrinsic ergodicity estimate: invariant measures whose free energy is within $Δ$ of the pressure are $O(\sqrtΔ)$-close to $μ_ϕ$ when tested against Hölder observables. As an application, every finite-word cylinder of positive $μ_ϕ$-measure yields a uniform pressure gap for the set of orbits avoiding that cylinder, leading in the entropy case to strict entropy and Hausdorff-dimension gaps.

math.DS↗

arXiv2025

Concentration inequalities and Transfer operators for supercritical Dyson models

The present paper extends the results on the ferromagnetic Dyson models from \cites{EFMV2024, JOP2025} to the near-critical and strongly interacting regimes. As part of our main result, we further establish Gaussian concentration bounds for the unique infinite-volume Gibbs measure throughout the entire supercritical regime.

math.PR↗

arXiv2025

Gibbs Properties of Equilibrium States

We consider the problem of equivalence of Gibbs states and equilibrium states for continuous potentials on full shift spaces $E^{\mathbb{Z}}$. Sinai, Bowen, Ruelle and others established equivalence under various assumptions on the potential $ϕ$. At the same time, it is known that every ergodic measure is an equilibrium state for some continuous potential. This means that the equivalence can occur only under some appropriate conditions on the potential function. In this paper, we identify the necessary and sufficient conditions for the equivalence.

math.DS↗

arXiv2024

On an extension of a theorem by Ruelle to long-range potentials

Ruelle's transfer operator plays an important role in understanding thermodynamic and probabilistic properties of dynamical systems. In this work, we develop a method of finding eigenfunctions of transfer operators based on comparing Gibbs measures on the half-line $\mathbb Z_+$ and the whole line $\Z$. For a rather broad class of potentials, including both the ferromagnetic and antiferromagnetic long-range Dyson potentials, we are able to establish the existence of integrable, but not necessarily continuous, eigenfunctions. For a subset thereof we prove that the eigenfunction is actually continuous.

math.DS↗

arXiv2024

Multifractal Formalism from Large Deviations

It has often been observed that the Multifractal Formalism and the Large Deviation Principles are intimately related. In fact, Multifractal Formalism was heuristically derived using the Large Deviations ideas. In numerous examples in which the multifractal results have been rigorously established, the corresponding Large Deviation results are valid as well. Moreover, the proofs of multifractal and large deviations are remarkably similar. The natural question then is whether under which conditions multifractal formalism can be deduced from the corresponding large deviations results. More specifically, given a sequence of random variables $\{ {X_n} \}_{n\in\N}$, satisfying a Large Deviation Principle, what can be said about the multifractal nature of the level sets $K_α=\{ω: \lim_{n} \frac{X_n(ω)}{n}=α\}$. Under some technical assumptions, we establish the upper and lower bounds for multifractal spectra in terms of the large deviation rate functions, and show that many known results of multifractal formalism are covered by our setup.

math.DS↗

arXiv2021

Description of the set of strictly regular quadratic bistochastic operators and examples

The present paper focuses on the dynamical systems of the quadratic bistochastic operators (QBO) on the standard simplex. In the paper, we show the character of connection of the dynamical systems of a bistochastic operator with the dynamical systems of the extreme bistochastic operators. In addition, we prove that almost all quadratic bistochastic operators are strictly regular and give a description of the strictly regular quadratic bistochastic operators in the convex polytope QBO. Furthermore, the density of the set of strictly regular QBO in the set of QBO is proved and nontrivial examples of strictly regular bistochastic operators are given.

math.DS↗