Tree-indexed Markov Chains Research Articles

The extremality of disordered phases for the mixed spin-(1,1/2) Ising model on a Cayley tree of arbitrary order

This paper continues the exploration of translation-invariant splitting Gibbs measures (TISGMs) within the framework of the Ising model with mixed spin-(1,1/2) (abbreviated as (1,1/2)-MSIM) on a Cayley tree of arbitrary order. Building upon our prior work (Akin and Mukhamedov 2022 J. Stat. Mech. 053204), where we extensively elucidated TISGMs and investigated the extremality of disordered phases employing a Markov chain indexed by a tree on a semi-infinite Cayley tree of order two, our current research extends this analysis. In the present study, we begin by constructing TISGMs and tree-indexed Markov chains tailored to the (1,1/2)-MSIM. Notably, we extend our scope to encompass Cayley trees of varying orders, thereby affording a comprehensive examination of the extremality characteristics inherent to disordered phases. Utilizing the Kesten–Stigum condition, we delve into the non-extremality aspects of disordered phases by scrutinizing the eigenvalues of the stochastic matrix associated with the (1,1/2)-MSIM on Cayley trees with order k ( k⩾3 ). This approach allows for a nuanced exploration of the model’s behavior and stability under different conditions.

The classification of disordered phases of mixed spin (2,1/2) Ising model and the chaoticity of the corresponding dynamical system

We study an Ising model having the mixed spins {±1/2} and {±2,±1,0} on Cayley tree of second-order. We construct the Gibbs measures corresponding to the model and classify the disordered phases associated to the Gibbs measures. Using the compatibility condition, we obtain the system of functional equations associated with the model. Contrary to the Ising model with two different neighbor interactions, we prove that for the given model, the phase transition phenomenon occurs in both the antiferromagnetic and antiferromagnetic regions. Stability analysis of the dynamic system associated with the model is performed at the obtained fixed point. By calculating the Lyapunov exponent numerically, we show that the corresponding dynamical system exhibits the chaotic behavior in some regions. We identify regions where the disordered phases are extreme by means of a tree-indexed Markov chain. We satisfy the Kesten–Stigum condition for non-extremality of the disordered phase according to the fixed point.

Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties

Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties

Phase transition for the Ising model with mixed spins on a Cayley tree

In the present paper, we consider the Ising model with mixed spin- (1, 1/2) on the second order Cayley tree. For this model, a construction of splitting Gibbs measures is given that allows us to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model exhibits three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, respectively, while the classical Ising model does not possess such Gibbs measures in the anti-ferromagnetic setting. It turns out, that like the classical Ising model, we can find a disordered Gibbs measure, therefore, its non-extremity and extremity are investigated by means of tree-indexed Markov chains.

A class of small deviation theorems for functionals of random fields on double Cayley tree in random environment

ABSTRACT In this paper, we introduce the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Meanwhile, we establish a class of small deviation theorems for functional of random fields on double Cayley tree in random environment. Finally, We prove the asymptotic equipartition property (AEP) for tree indexed Markov chains in Markovian environment with finite state space.

Hard-core and soft-core Widom–Rowlinson models on Cayley trees

We consider both hard-core and soft-core Widom–Rowlinson models with spin values on a Cayley tree of order and we are interested in the Gibbs measures of the models. The models depend on three parameters: the order k of the tree, describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and describing the intensity for particles. The hard-core Widom–Rowlinson model corresponds to the case .For the binary tree k = 2, and for k = 3 we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invariant Gibbs measures (TISGMs) which are tree-indexed Markov chains). We also give the exact form of the corresponding critical curves in parameter space. For higher values of k we give an explicit sufficient bound ensuring non-uniqueness which we conjecture to be the exact curve. Moreover, for the antiferromagnetic model we explicitly give two critical curves , , and prove that on these curves there are exactly two TISGMs; between these curves there are exactly three TISGMs; otherwise there exists a unique TISGM. Some periodic and non-periodic SGMs are also constructed in the ferromagnetic model.

THE SHANNON–MCMILLAN THEOREM FOR MARKOV CHAINS INDEXED BY A CAYLEY TREE IN RANDOM ENVIRONMENT

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

Conditional entropy, entropy density, and strong law of large numbers for generalized controlled tree-indexed Markov chains

ABSTRACTIn this paper, we first introduces a tree model without degree boundedness restriction namely generalized controlled tree T, which is an extension of some known tree models, such as homogeneous tree model, uniformly bounded degree tree model, controlled tree model, etc. Then some limit properties including strong law of large numbers for generalized controlled tree-indexed non homogeneous Markov chain are obtained. Finally, we establish some entropy density properties, monotonicity of conditional entropy, and entropy properties for generalized controlled tree-indexed Markov chains.

Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree

We continue our study of the full set of translation-invariant splitting Gibbs measures TISGMs, translation-invariant tree-indexed Markov chains for the q-state Potts model on a Cayley tree. In our previous work Kulske et al., J Stat Phys 156 2014, 189-200 we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2q-1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is non-extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2q-1+q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 636-678, 2017

The definition of tree-indexed Markov chains in random environment and their existence

ABSTRACTThis paper gives the definition of tree-indexed Markov chains in random environment with discrete state space, and then studies some equivalent theorems of tree-indexed Markov chains in random environment. Finally, we give the equivalence on tree-indexed Markov chains in Markov environment and double Markov chains indexed by a tree.

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity $$\lambda >0$$?>0 on the Cayley tree of order $$k\ge 1$$k?1. There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases $$\forall \lambda >0$$??>0 and $$k\ge 1$$k?1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at $$k=2$$k=2 if $$\lambda 9/4$$?>9/4 (resp. $$\lambda >1$$?>1), there exist three TISGMs. In this paper we generalize the result (ii) for any $$k\ge 2$$k?2, i.e., for hinge and wand cases we find the exact critical value $$\lambda _\mathrm{cr}(k)$$?cr(k) with properties mentioned in (ii). Moreover, we find some regions for the $$\lambda $$? parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.

Description of the Translation-Invariant Splitting Gibbs Measures for the Potts Model on a Cayley Tree

For the $q$-state Potts model on a Cayley tree of order $k\geq 2$ it is well-known that at sufficiently low temperatures there are at least $q+1$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is $2^{q}-1$. We prove that there are $[q/2]$ (where $[a]$ is the integer part of $a$) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs for each intermediate temperature. For the binary tree we give explicit formulae for the critical temperatures and the possible TISGMs. While we show that these measures are never convex combinations of each other, the question which of these measures are extremals in the set of all Gibbs measures will be treated in future work.

Tracer: Taming Anomalous Events with CRFID Tags for Trajectory Management

Mitigating anomalies are crucial for trajectory management in logistics and supply chain systems. Among variant devices for trace detection, computational radio frequency identification (CRFID) tags are promising to draw precise trajectory from the data reported by their accelerometers. However, full coverage of the processing flow using RFID readers is usually cost inefficient, sometimes impractical. In this paper, we propose to employ CRFID tags as tagging devices and develop a working system, Tracer, for precise trajectory detection. Instead of covering the entire processing area, Tracer only deploys RFID readers in essential regions to detect the mishandling, loss, and other abnormal states of items. We design a tree-indexed Markov chain framework, which leverages statistical methods to enable fine-grained and dynamic trajectory management. Results from a preliminarily deployment on a real baggage handling system and trace-driven simulations demonstrate that Tracer is effective to detect the anomalous events with low cost and high accuracy.

Statistical anomaly detection with sensor networks

We seek to detect statistically significant temporal or spatial changes in either the underlying process the sensor network is monitoring or in the network operation itself. These changes may point to faults, adversarial threats, misbehavior, or other anomalies that require intervention. To that end, we introduce a new statistical anomaly detection framework that uses Markov models to characterize the “normal” behavior of the sensor network. We develop a series of Markov models, including tree-indexed Markov chains which can model its spatial structure. For each model, an anomaly-free probability law is estimated from past traces. We leverage large deviations techniques to develop optimal anomaly detection rules for each corresponding Markov model, assessing whether its most recent empirical measure is consistent with the anomaly-free probability law. A series of simulation results, some with real sensor data, validate the effectiveness of the proposed anomaly detection algorithms.

Random fractals and tree-indexed Markov chains

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot’s fractal percolation process.

Random Wavelet Series Based on a Tree-Indexed Markov Chain

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection.

Influence of fine-scale heterogeneity patterns on the large-scale behaviour of miscible transport in porous media

An extensive series of numerical simulations on two-dimensional flow and miscible transport are carried out. The purpose of performing these simulations is to study the influence of fine-scale heterogeneity patterns (i.e. horizontal laminations, cross-bedding at 45°and 135°) with short range and long range correlation structure on large-scale behaviour of miscible transport in porous media. Synthetic heterogeneous structures have been generated using newly developed stochastic techniques (e.g. the coupled Markov chain method) to model the subsurface formations in various configurations, using realistic characteristics, and the tree-indexed Markov chain method to merge these heterogeneities at various scales. The results of the simulation are compared with simulations on a reference model of heterogeneity where there is no fine-scale heterogeneity. The simulations show that the variation in the fine-scale heterogeneity inside the large-scale lithological units has considerable impact on concentration distribution and the spreading of miscible plumes. The combination of the two stochastic techniques (the coupled Markov chain model and the tree-indexed Markov chain) is a useful tool to study multi-scale transport behaviour in heterogeneous media. The fine-scale heterogeneity enhances the mixing process, but the definition of an asymptotic giga-scopic dispersion at field scale is still questionable.

Reconstruction on Trees: Beating the Second Eigenvalue

We consider a process in which information is transmitted from a given root node on a noisy -dary tree network T. We start with a uniform symbol taken from an alphabet \(\mathcal{A}\). Each edge of the tree is an independent copy of some channel (Markov chain) M, where M is irreducible and aperiodic on \(\mathcal{A}\). The goal is to reconstruct the symbol at the root from the symbols at the nth level of the tree. This model has been studied in information theory, genetics and statistical physics. The basic question is: is it possible to reconstruct (some information on)the root? In other words, does the probability of correct reconstruction tend to \(1 /{\mathcal{A}}\) as n →∞? It is known that reconstruction is possible if dλ22(M) > 1, where λ2(M) is the second eigenvalue of M. Moreover,in this case it is possible to reconstruct using a majority algorithm which ignores the location of the data at the boundary of the tree. When M is a symmetric binary channel, this threshold is sharp. In this paper we show that, both for the binary asymmetric channel and for the symmetric channel on many symbols, it is sometimes possible to reconstruct even when dλ22(M) < 1. This result indicates that, for many (maybe most) tree-indexed Markov chains, the location of the data on the boundary plays a crucial role in reconstruction problems.

Multi-Scale and Multi-Resolution Stochastic Modeling of Subsurface Heterogeneity by Tree-Indexed Markov Chains

A new methodology is proposed to handle multi-scale heterogeneous structures. It can be of importance in the field of hydrogeology and for petroleum engineers who are interested in characterizing subsurface heterogeneity at various scales. The framework of this methodology is based on a coarse to fine scale representation of the heterogeneous structures on trees. Different depths in the tree correspond to different spatial scales in representing the heterogeneous structures on trees. On these trees a Markov chain is used to describe scale to scale transitions and to account for the uncertainty in the stochastically generated images. We focus in this work on the description and application of the methodology to synthetic data that are geologically realistic. The methodology is flexible. Conditioning on field data and measurements is straightforward. Non-stationary and stationary fields, compound and nested structures can be addressed.