A Constructive Description of Ground States and Gibbs Measures for Ising Model with Two-Step Interactions on Cayley Tree

We consider the Ising model with (competing) two-step interactions and spin values ± 1, on a Cayley tree of order k ≥ 1. We constructively describe ground states and verify the Peierls condition for the model. We define notion of a contour for the model on the Cayley tree. Using a contour argument we show the existence of two different Gibbs measures.     

Competing Ising interactions and chaotic glass-like behaviour on a Cayley tree

For the Ising model on a Cayley tree with competing nearest neighbour coupling J and next nearest neighbour coupling J′, we find in addition to the expected paramagnetic, ferromagnetic and antiferromagnetic phases, an intermediate range of J′/J < 0 values where the local magnetization has chaotic oscillatory glass-like behaviour.     

Fractal Failure of Quasilocality for a Majority Rule Transformation on a Tree

We provide a transformation of the Ising model on a Cayley tree leading to non-Gibbsianness at any temperature, i.e. even within the uniqueness regime. We also introduce a new type of pathologies of renormalized Gibbs measures, called the fractal failure of quasilocality, and exhibit a concrete example.     

Continuous order phase transitions on a decorated Cayley tree of connectivity two

The treatment of the Ising model on a Cayley tree given by Müller-Hartmann and Zittartz is extended in the case of connectivity two to a decorated tree containing additional bonds with an arbitrary coupling constant. The possibility of phase transitions is investigated and discussed. The positions of the singular surfaces, on which continuous order phase transitions take place, are examined as functions of coupling constants and external fields.     

Single Polygon Counting on Cayley Tree of Order 3

We showed that one form of generalized Catalan numbers is the solution to the problem of finding different connected component with finite vertices containing a fixed root for the semi-infinite Cayley tree of order 3. We give the formula for the full graph, Cayley tree of order 3 which is derived from the generalized Catalan numbers. Using ratios of Gamma functions, two upper bounds are given for problem defined on semi-infinite Cayley tree of order 3 as well as the full graph.     

New Phase Transitions of the Ising Model on Cayley Trees

We show that the nearest neighbor Ising model on the Cayley tree exhibits new temperature–driven phase transitions. These transitions occur at various inverse temperatures different from the critical one. They are characterised by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points. We also consider the model in the presence of an external field and compute the free energies of translation invariant and some alternating boundary conditions.     

s-State potts model on a Cayley tree

The Potts model is shown to exhibit a phase transition of continuous order on a Cayley tree. The leading nonanalytic part of the free energy ∣L∣ ^$\text{K}$_s involves a critical exponent $\text{K}$_s going from one to infinity as the coupling goes f rom infinity to K_BP, the Bethe-Peierls critical coupling.     

On Free Energies of the Ising Model on the Cayley Tree

We present, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). They include translation-invariant, periodic, Dobrushin-like b.c., as well as those corresponding to (recently discovered) weakly periodic Gibbs states. The weakly periodic measures are defined through a partition of the Cayley tree that induces a 4-edge-coloring on that tree. We compute the density of each color. We use these densities for computations of free energies corresponding to a weakly periodic b.c.     

Theory of the Ising model on a Cayley tree

The Ising model on a Cayley tree is known to exhibit a phase transition of continuous order. In this paper we present a complete and quantitative analysis of the leading singular term in the free energy which is associated with this phase transition. We have been able to solve this problem by considering the distribution of zeros of the partition function. The most interesting new feature in our results is a contribution to the free energy which performs singular oscillations as the magnetic field approaches zero.     

Characteristics and synchronization of time-delay systems driven by a common noise

We investigate the characteristics of time-delay systems in the presence of Gaussian noise. We show that the delay time embedded in the time series of time-delay system with constant delay cannot be estimated in the presence noise for appropriate values of noise intensity thereby forbidding any possibility of phase space reconstruction. We also demonstrate the existence of complete synchronization between two independent identical time-delay systems driven by a common noise without explicitly establishing any external coupling between them.     

The role of power law nonlinearity in the discrete nonlinear Schrödinger equation on the formation of stationary localized states in the Cayley tree

We study the formation of stationary localized states using the discrete nonlinear Schr?dinger equation in a Cayley tree with connectivity K. Two cases, namely, a dimeric power law nonlinear impurity and a fully nonlinear system are considered. We introduce a transformation which reduces the Cayley tree into an one dimensional chain with a bond defect. The hopping matrix element between the impurity sites is reduced by . The transformed system is also shown to yield tight binding Green's function of the Cayley tree. The dimeric ansatz is used to find the reduced Hamiltonian of the system. Stationary localized states are found from the fixed point equations of the Hamiltonian of the reduced dynamical system. We discuss the existence of different kinds of localized states. We have also analyzed the formation of localized states in one dimensional system with a bond defect and nonlinearity which does not correspond to a Cayley tree. Stability of the states is discussed and stability diagram is presented for few cases. In all cases the total phase diagram for localized states have been presented. Received: 18 September 1997 / Revised: 31 October and 17 november 1997 / Accepted: 19 November 1997     

Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.     

Blocking and Dimer Processes on the Cayley Tree

We show that the equilibrium distribution for the dimer process on the finite Cayley tree tends to a translation invariant limit as the size of the tree tends to infinity. The same is true for the blocking process except when there is a phase transition, in which case there are two limits, each a one-step translation of the other. We also find correlations for occupation probabilities.     

Lasing without inversion in circuit quantum electrodynamics

We study the photon generation in a transmission line oscillator coupled to a driven qubit in the presence of a dissipative electromagnetic environment. It has been demonstrated previously that a population inversion in the qubit can lead to a lasing state of the oscillator. Here we show that the circuit can also exhibit the effect of "lasing without inversion." It arises since the coupling to the dissipative environment enhances photon emission as compared to absorption, similar to the recoil effect predicted for atomic systems. While the recoil effect is very weak, and so far elusive, the effect described here should be observable with realistic circuits. We analyze the requirements for system parameters and environment.     

Weakly Periodic Gibbs Measures of the Ising Model on the Cayley Tree of Order Five and Six

For Ising model on the Cayley tree of order five and six we present new weakly periodic (non-periodic) Gibbs measures corresponding to normal subgroups of indices two in the group representation of the Cayley tree.     

Controlling the leader-laggard dynamics in delay-synchronized lasers

We study experimentally the collective dynamics of two delay-coupled semiconductor lasers. The lasers are coupled by mutual injection of their emitted light beams, at a distance for which coupling delay times are non-negligible. This system is known to exhibit lag synchronization, with one laser leading and the other one lagging the dynamics. Our setup is designed such that light travels along different paths in the two coupling directions, which allows independent control of the two coupling strengths. A comparison of unidirectional and bidirectional coupling reveals that the leader-laggard roles can be switched by acting upon the coupling architecture of the system. Additionally, numerical simulations show that a more extensive control of the network architecture can also lead to changes in the dynamics of the system. Finally, we discuss the relevance of these results for bidirectional chaotic communications.     

A Manifold of Pure Gibbs States of the Ising Model on a Cayley Tree

We study the Ising model on a Cayley tree. A wide class of new extreme Gibbs states is exhibited.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J₁) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.     

Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models

We give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontanoous symmetry breaking appears or disappears. The two systems are the hard-core model and the Widom–Rowlinson model, and both examples take place on a variation of the Cayley tree (Bethe lattice) devised by Schonmann and Tanaka. We obtain similar, though less constructive, examples of nonmonotonicity via certain local modifications of any graph, e.g., the square lattice, which is known to have a critical point for either model. En route we discuss the critical behavior of the Widom–Rowlinson model on the ordinary Cayley tree. Some results about monotonicity of the phase transition phenomenon relative to graph structure are also given.