Bethe Lattice Research Articles

Inhomogeneous percolation on the Bethe lattice with critical exponents and its application

This manuscript explores the critical exponents for three-dimensional irregular Bethe lattice (TDIBL) and their closer association with real-life applications. In the current investigation, we evaluate the CEs for TDIBL using several percolating variables, including the cluster size distribution (CSD), percolating probability (PP), mean cluster size (MCS), characteristic cluster size (CCS), and characteristic generation size (CGS). The CEs for TDIBL findings illustrate that the percolation model accomplishes enormous universal classes and is connected with a complicated fractal and resizing structure of underlying geometric characteristics, which will contribute to the system’s mean degree in the percolating method. Additionally, sensitivity analysis and numerical simulation will further enhance our appraisal of the percolating procedure graphically. Furthermore, we address the novel coronavirus 2019 (COVID-19) transmission behavior in the database using our methodology. Utilizing this data type to develop predictions by conveying policy-using courtesy groups with varying infection and recovery probabilities is also feasible. We acknowledge that our revolutionary endeavor will be intriguing, and we will inquire about scientific researchers collaborating with various percolation methods of investigation.

Quantitative behavior of (1,1/2)-MSIM on a Cayley tree

In this paper, using the self-similarity technique, we establish the Gibbs measures related to the Ising model having mixed spin (1,1/2) (shortly, (1,1/2)-MSIM) on the semi-infinite Cayley tree (CT) and rigorously investigate the phase transition phenomena for (1,1/2)-MSIM on the CT by looking at the non-uniqueness of the fixed points for the relevant dynamic system. We demonstrate that the model displays chaotic behavior in some regions via the Lyapunov exponent. We compute the thermal average spin for the model. We present some exact formulas to calculate the free energy and entropy associated with (1,1/2)-MSIM on the second-order CT by employing the cavity technique.

Periodic ground states for the mixed spin ising model with competing interactions on a Cayley tree

Periodic ground states for the mixed spin ising model with competing interactions on a Cayley tree

Smoothness of Integrated Density of States of the Anderson Model on Bethe Lattice in High Disorder

In this work we consider the Anderson model on Bethe lattice and prove that the integrated density of states (IDS) is as smooth as the single site distribution (SSD), in high disorder.

The Blume-Emery-Griffiths Model with Competing Interactions on the Bethe Lattice

The Blume-Emery-Griffiths Model with Competing Interactions on the Bethe Lattice

Calculation of some thermodynamic quantities for the Ising model on a [formula omitted]th order Cayley tree

This paper focuses on the Ising model with first- and second-neighbor interactions on a kth order Cayley tree (k≥2). We generalize the Ising-Vannimenus model’s Gibbs measures on a Cayley tree with any order by means of Kolmogorov consistency condition. We classify the fixed points of the dynamical system associated with the model under given conditions. We demonstrate how the chaoticity of the corresponding system is influenced by the Cayley tree’s order. Depending on the number k, we obtain a new formula to calculate the free energy of the model on Cayley tree of any order under given boundary conditions. We determine the system’s entropy, which is the derivative of the free energy function with respect to temperature T.

P$$ p $$‐Adic phase transitions for countable state Potts model

The statistical mechanical models with finitely many spin values have several applications in many areas of natural science. One of the most studied models is the Potts model, which has a rich structure of physical phenomena. However, countable analog of the mentioned model has not been deeply studied yet. In the present paper, we are going to construct generalized ‐adic Gibbs measures (for the countable state ‐adic Potts model on a Cayley tree) by investigating non‐linear ‐adic dynamical systems on the sequence spaces. The provided study offers the existence of the strong phase transition for the model.

Potts model on a Cayley tree: a new class of Gibbs measures

Potts model on a Cayley tree: a new class of Gibbs measures

Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree

In this paper, we study the HC-model with a countable set $$\mathbb Z$$ of spin values on a Cayley tree of order $$k\ge 2$$ . This model is defined by a countable set of parameters (that is, the activity function $$\lambda _i>0$$ , $$i\in \mathbb Z$$ ). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:

Degree distributions in recursive trees with fitnesses

AbstractWe study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function, which is a function of its weight and degree, and connects to $\ell$ new-coming vertices. Under a certain technical assumption, applying the theory of Crump–Mode–Jagers branching processes, we derive formulas for the limiting distributions of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we rigorously prove observations of Bianconi related to the evolving Cayley tree (Phys. Rev. E66, paper no. 036116, 2002). We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call ‘generalised preferential attachment with fitness’. We show that this model can exhibit condensation, where a positive proportion of edges accumulates around vertices with maximal weight, or, more drastically, can have a degenerate limiting degree distribution, where the entire proportion of edges accumulates around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process.

Destruction of Anderson localization by subquadratic nonlinearity

It is shown based on a mapping procedure onto a Cayley tree that a subquadratic nonlinearity destroys Anderson localization of waves in nonlinear Schrödinger lattices with randomness, if the exponent of the nonlinearity satisfies , giving rise to unlimited subdiffusive spreading of an initially localized wave packet along the lattice. The focus on subquadratic nonlinearity is intended to amend and generalize the special case s = 1, considered previously, by offering a more comprehensive picture of dynamics. A transport model characterizing the spreading process is obtained in terms of a bifractional diffusion equation involving both long-time trappings of unstable modes on finite clusters and their long-haul jumps in wave number space consistent with Lévy flights. The origin of the flights is associated with self-intersections of the higher-order Cayley trees with odd coordination numbers z > 3 leading to degenerate states.

Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree

Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree

The description of generalized translation-invariant $$p$$-adic Gibbs measures for the Potts model on the Cayley tree of order three

The description of generalized translation-invariant $$p$$-adic Gibbs measures for the Potts model on the Cayley tree of order three

Degree-Based Entropy of Some Classes of Networks

A topological index is a number that is connected to a chemical composition in order to correlate a substance’s chemical makeup with different physical characteristics, chemical reactivity, or biological activity. It is common to model drugs and other chemical substances as different forms, trees, and graphs. Certain physico-chemical features of chemical substances correlate better with degree-based topological invariants. Predictions concerning the dynamics of the continuing pandemic may be made with the use of the graphic theoretical approaches given here. In Networks, the degree entropy of the epidemic and related trees was computed. It highlights the essay’s originality while also implying that this piece has improved upon prior literature-based realizations. In this paper, we study an important degree-based invariant known as the inverse sum indeg invariant for a variety of graphs of biological interest networks, including the corona product of some interesting classes of graphs and the pandemic tree network, curtain tree network, and Cayley tree network. We also examine the inverse sum indeg invariant features for the molecular graphs that represent the molecules in the bicyclic chemical graphs.

A novel method for the 3D inhomogeneous percolation on the Lattice with its application

This article anticipates the puzzle of acquiring a three-dimensional inhomogeneous site percolation on an irregular Bethe Lattice (TDIBL). In this work, we explore a TDIBL with the intent to realize the right tradeoff among different percolating variates, namely, cluster size distribution (CSD), critical occupation probability (COP), percolating probability (PP), and mean cluster size (MCS). The variates results are acquired using the generating function (GF) and generalized recursive approaches (GRA). The findings revealed that, for inhomogeneous site percolation (ISP) on the proposed model, the high fraction of probabilities (occupation and distribution probabilities) boosts the intensity of the process, which will enlarge the mean degree of the system in the percolation process. Moreover, numerical simulation and sensitivity analysis will also enhance our understanding of the percolating process via 2D and 3D. Their corresponding shape profiles of earned findings are drawn to perceive their dynamics amongst the options of entailed parameters. Furthermore, exhausting the above scheme, we discuss the transmission behavior of the novel corona-virus 2019 (COVID-19) and present particular disease-control schemes courtesy of groups with numerous infection probabilities and find the effect according to the age distribution. We recognize that this endeavor is timely, that it will be about curiosity, and that it will include scientists working with diverse percolation approaches.

Linked cluster expansion on trees.

The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.

Gibbs measures for the Potts model with a countable set of spin values on a Cayley tree

Gibbs measures for the Potts model with a countable set of spin values on a Cayley tree

Gibbs measures of an Ising-Vannimenus Model with one-level competing interactions on 4th order Cayley tree

In this paper, we study an Ising-Vannimenus model on a fourth-order semi-infinite Cayley tree with one-level next-nearest-neighbor interactions. We construct the set of translation-invariant Gibbs measures corresponding to the model satisfying Kolmogorov consistency conditions. We prove the existence of the phase transition under specific conditions on the coupling constants and temperature. Numerical examples are presented to clarify the role of the added one-level interactions and results.

Fixed points of an infinite-dimensional operator related to Gibbs measures

We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters $\lambda_{i}>0$, $a_{ij}\in \{0,1\}$, $ i,j \in \mathbb{N}$. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs measure for the given HC-model.

Dynamical Gibbs–non-Gibbs transitions in Widom–Rowlinson models on trees

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order $d$ (which has $d + 1$ nearest neighbours), depending on repulsion strength $\beta$ between particles of different signs and on an activity parameter $\lambda$ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength $\beta$. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any $d \geq 2$. In our second and main result, we also show that for large $\beta$ and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity $\lambda$ increases, assuming that $d \geq 4$. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.