arXiv subjects

Farrukh Mukhamedov

Publications and source records attributed to Farrukh Mukhamedov.

At least 19 recordsLinked to original sources

arXiv2026

Branching Gaps, Coexistence, and Boundary Influence for the Ising Model on Spherically Symmetric Trees

In this paper, we study the ferromagnetic nearest-neighbor Ising model on rooted spherically symmetric trees with one or $q$ children per vertex, according to a prescribed sequence of branching levels. Unbounded runs of one-child levels force uniqueness at every nonzero homogeneous field and every finite inverse temperature. We quantify this effect through the locations of sufficiently long nonbranching stretches. Conversely, uniqueness need not entail any uniform rate: for each prescribed vanishing sequence, we construct a density-one tree whose finite-volume boundary influence decays more slowly along a subsequence. The tree retains branching number $q$, although its nonzero-field coexistence region disappears. For eventually bounded gaps, we prove an optimal uniform comparison with the periodic tree having the largest allowed gap and obtain an explicit interval of coexistence fields. Thus bounded gaps are equivalent, within this family, to coexistence at some finite temperature and nonzero field. For eventually periodic gaps, Gibbs uniqueness is equivalent to fixed-point uniqueness of the full-period return map. A negative Schwarzian derivative yields a complete classification: in the supercritical regime the map has three fixed points inside a closed coexistence interval, two at its endpoints, and one outside. The nonzero endpoints are automatically nondegenerate folds, with square-root splitting of the merging branches.

math-ph↗

arXiv2026

Quasi-Product States and Factor Types for the One-Dimensional Hard-Core Model

We study the von Neumann algebra generated by a quasi-product state associated with the one-dimensional hard-core lattice gas. If $κ>0$ denotes the activity (fugacity), the transfer-matrix construction gives the stationary transition matrix \[ P=\begin{pmatrix}p&q\\ 1&0\end{pmatrix}, \qquad p+q=1, \qquad κ=\frac{q}{p^2}. \] The stationary Markov measure on the hard-core path space induces a faithful diagonal state on the corresponding path $AF$-algebra. We give a state-preserving identification of its GNS von Neumann algebra with the Feldman--Moore algebra of the measured tail equivalence relation. The relation is hyperfinite and ergodic, and its nonzero ratio set is $κ^{\mathbb Z}$. Consequently, the generated algebra is the hyperfinite factor of type $\mathrm{II}_1$ when $κ=1$, and of type $\mathrm{III}_λ$, where $λ=\min\{κ,κ^{-1}\}$, when $κ\ne1$. The point $κ=1$ is not a thermodynamic phase transition: the pressure and particle density are analytic for all $κ>0$. We also describe the precise support-corner relation with the GNS algebra of the diagonal Gibbs state on the quasi-local spin algebra, and we determine the centralizer and the flow of weights.

math-ph↗

arXiv2026

Ballistic Speed and Potential-Theoretic Recurrence for Homogeneous Open Quantum Random Walks

Our study connects root-mean-square ballistic transport to potential-theoretic recurrence for finite-range open quantum random walks. In homogeneous walks with primitive local channels, finitely many simple periodic Fourier peripheral eigenvalues, no nonzero stationary Fourier mode, and a nondegenerate quadratic spectral term, we prove a periodic uniform local limit theorem and recover exponential finite-set return bounds for nonzero drift. The centered case yields strong Green-function asymptotics in at least three dimensions, as well as potential-kernel asymptotics in one and two dimensions. According to these assumptions, the RMS ballistic speed equals the drift norm, nonzero drift indicates transience, and centered walks are recurring in effective dimensions one and two but transitory in higher dimensions. The low-dimensional finding shows a recurrence of the origin projection in TOM. For reducible walks, an explicit harmonic $h$-transform converts each absorption component to an OQRW, providing a detailed breakdown of Green occupation potentials. The squared RMS speed is the absorption-weighted mean of squared component drifts, but the reduced drift-dimension classification also needs specific component return estimations. A centered noncommuting family validates the fundamental spectral assumptions in all dimensions and provides explicit potential constants. Exact finite traps and sparse reflecting barriers provide a complementary nonhomogeneous method for zero speed and TOM recurrence.

math.FA↗

arXiv2026

Sparse empirical measures for non-statistical interval maps with two neutral fixed points

We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $β=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $ν_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\in\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_ϕ(n\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\tfrac12(δ_{-1}+δ_1)$.

math.DS↗

arXiv2026

Full-measure historic behavior and emergence in simplex dynamics

We give a finite-dimensional boundary-residence mechanism for Takens-type historic behavior. For a class of continuous self-maps of a simplex, a multiplicative Lyapunov drift toward the boundary, combined with cyclic residence near separated boundary regions, forces every non-fixed interior orbit to have non-convergent Cesàro averages and non-convergent empirical measures. Thus the historic set has full relative Lebesgue measure. The mechanism is independent of hyperbolic specification, wandering domains and positive entropy: in fact, for the maps considered here the topological entropy is carried entirely by the boundary restriction. The non-convergence persists under all positive powers and, more generally, under multiplicatively thick sampling times. The abstract criterion applies to Stein--Ulam and Lotka--Volterra type stochastic operators once the model-specific cyclic coding and logarithmic residence estimates are verified. In explicit cyclic-collar models we identify the full weak-* accumulation set of empirical measures; this accumulation set is a non-trivial circle of probability measures and yields polynomial pointwise emergence of order $\varepsilon^{-1}$. We also obtain examples with positive-dimensional omega-limit sets by combining the mechanism with a Barański--Misiurewicz type boundary-tracing construction.

math.DS↗

arXiv2026

Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants

The $p$-adic method for Dynamical Mordell--Lang often reduces a residue class of an orbit to the zero set of a locally analytic interpolating function. This paper assumes the standard interpolation, Strassmann, Mahler, and Weierstrass tools, and studies the effective local zero-bound problem that remains after interpolation: certifying the Strassmann index of the resulting one-variable analytic function. We give finite certificates for this index, including finite-data, finite-precision, refined-tail, adaptive residue-class, one-shot, and first-order escape criteria. Since the Strassmann index is a rigorous upper bound for zeros in $\Zp$ and, through Weierstrass preparation, a root count on the closed disc over $\Cp$, these certificates give checkable stopping criteria for local orbit-intersection computations. A residue-class zooming principle replaces a congruence class of times by the iterate $f^{p^h}$, gaining $h$ additional powers of $p$ in the certificate tails. We also introduce an arc-ideal viewpoint for target varieties defined by several equations, replacing a chosen hypersurface bound by the one-variable gcd of all defining equations along the interpolated orbit. In dimension one, the method identifies the certified Strassmann index with the corresponding local Weierstrass root count in the orbit ball. Applications include certified bounds for intersections of non-fixed power-map orbits with finite target sets, and root-of-unity avoidance for maps tangent to the identity at torsion units.

math.DS↗

arXiv2026

Quantum Markov Chains for an Asymmetric Mixed Ising-XY Model on a Cayley Tree

We study a mixed quantum Ising-$XY$ model on the semi-infinite rooted Cayley tree of order two. For every vertex $u$, the edge $\langle u,(u,1)\rangle$ carries an $XY$ interaction and the edge $\langle u,(u,2)\rangle$ carries an Ising interaction. Using the compatibility criterion for tree-indexed quantum Markov chains and consistently working with the normalized trace, we derive the translation-invariant boundary equation and compute explicitly the associated local transfer operator, namely the one-step partial-trace map which propagates successor boundary data to the parent vertex. We prove that the boundary equation has a unique positive translation-invariant solution for all $J_I,J_{XY}\in\mathbb R$ and $β>0$. Hence the model admits a unique translation-invariant quantum Markov chain generated by a positive translation-invariant boundary condition. We also show that the reduced boundary-law dynamics, i.e. the induced finite-dimensional recursion for the boundary-law parameters, has no admissible periodic points of period greater than one and compute the local two-site entanglement on the natural three-site cluster of the tree.

math-ph↗

arXiv2026

Local Rota--Baxter operators of weight zero on nilpotent evolution algebras with maximal nilindex

Let $E$ be an $n$-dimensional nilpotent evolution algebra of maximal nilindex over a field of characteristic zero. The Rota--Baxter operators of weights zero and one on such algebras were recently classified. In this paper we investigate the local analogue of the weight-zero case. We prove that every local Rota--Baxter operator of weight zero has a rigid diagonal part: the nonzero diagonal coefficients, when they occur, begin after the obstruction index determined by the structural matrix of $E$ and form a final geometric string governed by one scalar. In contrast, the last-column coefficients are arbitrary. This gives an explicit description of the class $\LRB_0(E)$ and shows that it is generally strictly larger than $\RB_0(\E)$. We also prove that the corresponding $s$-local notion produces no new operators. The classification is further interpreted as a finite union of quasi-affine strata, yielding a dimension comparison between $\LRB_0(E)$ and $\RB_0(E)$. Finally, we study ordinary weight-zero Rota--Baxter operators commuting with derivations and automorphisms and describe the resulting conditions in terms of the directed graph associated with $E$.

math.RA↗

arXiv2026

Quantum Viterbi Algorithm

We introduce a quantum Viterbi decoding algorithm for hidden quantum Markov models (HQMMs) motivated by quantum information processing and quantum algorithms. Given a finite sequence of measurement outcomes, the algorithm identifies hidden quantum trajectories that maximize a joint decoding functional, serving as a genuine quantum analogue of the classical Viterbi score. Unlike classical hidden Markov models, where decoding optimizes over a finite discrete state space, our method performs optimization over a continuous manifold of pure quantum effects, thereby exploiting coherent superpositions in the hidden memory. We prove a strict quantum advantage: coherent hidden trajectories can achieve decoding scores that strictly exceed any classical strategy constrained to diagonal (commuting) effects, even when both models share the same observed statistics. These results position quantum Viterbi decoding as a concrete quantum algorithmic primitive for sequential decision-making, with direct applications to quantum memories, quantum communication with memory, and near-term quantum machine learning on NISQ devices.

quant-ph↗

arXiv2026

Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators

We study weighted Walsh--Carleson maximal operators arising from dyadic martingale transforms associated with Walsh--Fourier partial sums. For weights satisfying a uniform dyadic variation condition and a uniform bound at the top dyadic scale, we prove weak type~$(1,1)$ estimates for the corresponding maximal operators along subsequences. We also give divergence criteria in terms of the behavior of the weights near the top dyadic scale and, under suitable admissibility assumptions, relate these criteria to explicit ratio conditions. As applications, we obtain results on matrix transforms of Walsh--Fourier partial sums, including de la Vallée Poussin means, Cesàro means with varying parameters, Nörlund logarithmic means, and general Nörlund means. In particular, we prove a Walsh--Paley analogue of the Leindler--Tandori theorem and establish everywhere divergence results for several summability methods.

math.CA↗

arXiv2026

Interpolating between positive, Schwarz, and completely positive evolution for d-level systems

We study a class of quantum dynamical maps for d-level systems that interpolate between positive, Schwarz, and completely positive evolutions. Our approach is based on a geometric analysis of the parameter space, which reveals the structure of regions corresponding to different positivity classes and their boundaries. We show that dynamical trajectories naturally move across these regions, providing a clear geometric interpretation of transitions between Markovian and non-Markovian regimes. It is shown that within presented class the evolution becomes eventually entanglement breaking. This analysis highlights the role of divisibility and eternally non-Markovian evolution.

quant-ph↗

arXiv2026

Constructing $k$-Kadison-Schwarz maps

We study k Kadison Schwarz (k KS) mappings on matrix algebras and derive explicit conditions ensuring the k KS property for two classes of maps parameterized by a single k-positive map.

math.FA↗

arXiv2026

Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree

We consider an Ising model on the Cayley tree $Γ_k$ of arbitrary order $k\ge1$ with three spin species of values $(\tfrac12,1,\tfrac32)$ distributed deterministically with period three along the generations. Within the framework of splitting Gibbs measures, we derive the exact boundary-law compatibility equations and characterize translation-invariant splitting Gibbs measures (TISGMs) via a finite system of algebraic relations. In the ferromagnetic regime $J>0$, writing $θ=\exp(βJ/2)$, we further reduce the translation-invariant problem to a one-dimensional scalar fixed-point equation $x=f(x,θ,k)$ for a rational map $f$. We show that $f$ is strictly increasing and obtain an explicit sufficient condition for phase coexistence: if $s_k(θ)=f'(1,θ,k)-1>0$, then $x=f(x,θ,k)$ admits at least three distinct positive solutions, yielding at least three distinct TISGMs and hence a phase transition driven by the periodic inhomogeneity of the spin structure. For the binary tree $k=2$ we exploit attractiveness to construct plus and minus Gibbs measures as weak limits with extremal boundary conditions, prove that they are TISGMs corresponding to the minimal and maximal fixed points of $f(\cdot,θ,2)$, and show that they are the minimal and maximal Gibbs measures in the natural stochastic order. Finally, we construct the tree-indexed Markov chain associated with a TISGM and apply the Kesten--Stigum criterion to the disordered TISGM, identifying nonempty parameter regions where this measure is non-extremal and reconstruction occurs.

math.PR↗

arXiv2026

Rota-Baxter operators of nilpotent evolution algebras with maximal nilindex

Nilpotent evolution algebras of maximal nilindex admit a natural basis in which the structure matrix is strictly upper triangular. In this paper we classify Rota{Baxter operators of weights zero and one on such algebras. We prove that every Rota{Baxter operator is upper triangular with respect to a natural basis. For weight zero, a strong rigidity phenomenon occurs: the operators are diagonal up to possible perturbations supported in the last basis vector. For weight one, a richer structure appears, including both triangular and non-triangular families, with the diagonal entries governed by a rational recurrence relation. Our results provide a complete description of Rota{Baxter operators on nilpotent evolution algebras of maximal nilindex.

math.RA↗

arXiv2025

p-adic dynamical systems for $p$-adic $λ$-models on the Cayley trees

We review and propose to use of associated dynamical system to explore the phase transition phenomena in $p$-adic statistical mechanics setting, by means of the renormalization techniques. Main focus of the paper is the $p$-adic $λ$-model on the Cayley tree. We study generalized $p$-adic quasi Gibbs measures for the $λ$-model. These measures associate recurrence equations that determine dynamical systems for the model. A main point is to verify and confirm that the indicated predictions from a dynamical system point of view are indeed true.

math-ph↗

arXiv2025

Unital Kadison-Schwarz Maps

Quantum entanglement is an important phenomenon in quantum information theory. To detect entanglement theoretically, positive but not completely positive maps are used. The Kadison-Schwarz (KS) inequality interpolates between positivity and complete positivity. KS maps may be key to understanding and detecting entanglement. We provide a description of a subset of KS maps on $M_2(\mathbb{C})$ that are unital. This allows for the classification of a wider class of positive maps than the well known bistochastic maps. We derive the conditions for a unital map to be a KS map, and provide non-trivial examples of such a map.

quant-ph↗

arXiv2024

Limits of sequences of operators associated with Walsh System

The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by Gát and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.

math.AP↗

arXiv2024

Limits of sequence of Tensor Product Operators associated with the Walsh-Paley system

It is well-known that to establish the almost everywhere convergence of a sequence of operators on $L_1$-space, it is sufficient to obtain a weak $(1,1)$-type inequality for the maximal operator corresponding to the sequence of operators. However, in practical applications, the establishment of the mentioned inequality for the maximal operators is very tricky and difficult job. In the present paper, the main aim is a novel outlook at above mentioned inequality for the tensor product of two weighted one-dimensional Walsh-Fourier series. Namely, our main idea is naturally to consider uniformly boundedness conditions for the sequences which imply the weak type estimation for the maximal operator of the tensor product. More precisely, we are going to establish weak type of inequality for the tensor product of two dimensional maximal operators while having the uniform boundedness of the corresponding one dimensional operators. Moreover, the convergence of the tensor product of operators is established at the two dimensional Walsh-Lebesgue points.

math.AP↗