Random walks on the Bethe lattice

We obtain random walk statistics for a nearest-neighbor (Pólya) walk on a Bethe lattice (infinite Cayley tree) of coordination numberz, and show how a random walk problem for a particular inhomogeneous Bethe lattice may be solved exactly. We question the common assertion that the Bethe lattice is an infinite-dimensional system.Supported in part by the U.S. Department of Energy.     

Reentrant phase transitions and multicompensation points in the mixed-spin Ising ferrimagnet on a decorated Bethe lattice

A mixed-spin Ising model on a decorated Bethe lattice is rigorously solved by combining the decoration–iteration transformation with the method of exact recursion relations. Exact results for critical lines, compensation temperatures, total and sublattice magnetizations are obtained from a precise mapping relationship with the corresponding spin-1/2 Ising model on a simple (undecorated) Bethe lattice. The effect of next-nearest-neighbour interaction and single-ion anisotropy on magnetic properties of the ferrimagnetic model is investigated in particular. It is shown that the total magnetization may exhibit multicompensation phenomenon and the critical temperature vs. the single-ion anisotropy dependence basically changes with the coordination number of the underlying Bethe lattice. The possibility of observing reentrant phase transitions is related to a high enough coordination number of the underlying Bethe lattice.     

Solution of lattice gas models in the generalized ensemble on the Bethe lattice

We present the exact Bethe lattice solution for a lattice gas Potts model defined in the generalized ensemble which was previously studied in elucidating the anomalous thermodynamic properties of water. For this model the locus of density maxima (TMD), the locus of isothermal compressibility extrema, (TEC), the spinodal curve, and the percolation curve for four hydrogen bonded molecules are calculated using the Bethe lattice solution. The results confirm qualitative relationships between the TMD, the TEC, and the percolation curve obtained previously from a mean field calculation.     

The Bethe lattice spin glass revisited

So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization. Received 27 September 2000     

Random walk on a disordered directed Bethe lattice

The random walk of a particle on a directed Bethe lattice of constant coordinanceZ is examined in the case of random hopping rates. As a result, the higher the coordinance, the narrower the regions of anomalous drift and diffusion. The annealed and quenched mean square dispersions are calculated in all dynamical phases. In opposition to the one-dimensional (Z=2) case, the annealed and quenched mean quadratic dispersions are shown to be identical in all phases.We shall employ indifferently the expressions Bethe lattice or infinite Cayley tree to denote an infinite ramified lattice of constant coordinanceZ.^{(4, 5)}     

The Bethe approximation for lattice gauge theories

The Bethe approximation is defined for general lattice gauge theories. It amounts to solving the model on an infinite Cayley lattice of cubes. The approximation is tested on the 4-d Z₄ model, where it is shown to reproduce accurately most of the phase diagram. It also suggests which mass vanishes in the Coulomb phase.     

A new formulation for the bethe approach to lattice statistics

An improved mathematical treatment is given for the approximate statistics of the Ising problem based on configurations of local Bethe clusters. The essential step in the development is to write the configuration probability for the Bethe cluster in terms of the probabilities for subclusters. Exact statistical mechanical results are used to interrelate the probabilities. The technique is illustrated first for a simple ferromagnetic lattice and then applied to the order-disorder problems of binary and ternary alloys on the face centered cubic lattice. In each ease the final equations which need solution are derived by strictly algebraic means. The improvement in mathematical treatment is such that these equations are identical with high order approximations of the very different cluster variation method.     

Infinite divisibility of a bethe lattice ising model

It is shown that the probability distribution for the infinite-volume, free-boundary-condition Ising ferromagnet on the Bethe lattice under zero external field is infinitely divisible with respect to the group operation of pointwise multiplication of spin variables.     

Statistical mechanics of a Bethe lattice

The statistical mechanics of a spin-one Bethe lattice is discussed. The exact expression for Curie temperature is derived using a Bethe-Peierls approximations. The effect of the inclusion of a single-spin anisotropy to each Ising model of the lattice is also studied. It is seen that under certain situations a first-order phase transition appears.     

Magnetization curves of a spin-one Bethe lattice

An exact calculation of the spontaneous magnetization of a spin-one Ising model with the inclusion of biquadratic exchange and single ion anisotropy on the Bethe lattice is presented. The magnetization curves exhibit some unusual features as the relative strength of the single ion anisotropy is varied.     

Curie temperature of a spin-one bethe lattice

The exact expression for the Curie temperature of a spin-1 Bethe lattice with dipolar and quadrupolar interactions is derived and the nature of variation of the Curie temperature with quadrupolar interaction constant is studied.     

Total and surface density of states on the Bethe lattice

A novel approach for calculating the density of surface states in a disordered system is presented. It is shown that the node counting method for calculating the energy spectrum of a system holds for the Bethe lattice. An exact method for calculating the total density of states of a disordered Bethe system terminating on a closed surface, as well as the contribution of the bulk and surface states to it is then developed and applied to the cases of Lorentzian and Gaussian randomness. The results are discussed in connection to relevant material from the literature.     

A spin-one Ising model on the Bethe lattice

We study the statistical mechanics of spin-one Ising models on the Bethe lattice assuming that the spins interact by dipolar and quadrupolar interactions. An exact calculation of the properties of the system is performed on the basis of the general formulation of Morita. An exact expression for the Curie temperature is derived and the results are found to be in agreement with those of Obokata and Oguchi who utilized a generalized Bethe approximation to a spin-one Ising system. The nature of variation of the Curie temperature with respect to the change of quadrupolar exchange is discussed for various coordination numbers and the results agree qualitatively with the earlier works. The temperature variation of both dipolar and quadrupolar moments is studied.     

Two-spin correlation function of spin 12 Ising model on a bethe lattice

The two-spin correlation function of the Ising model on a Bethe lattice as a function of temperature and magnetic field. It is found that the coherence length is finite at finite temperatures.     

A simple approximation for Ising spin models

A simple method is introduced for Ising spin systems based on the correlated molecular-field theory. Here, we consider an interacting triple in the mean field of its surrounding lattice points taking into account the product of three spins in the cluster. The present results for Curie temperature are compared to the mean-field theory(MFT), Bethe–Peierls–Weiss(BPW) approach, and self-consistent correlated field(SCCF) approximation, respectively. A procedure of triangular clusters is investigated that yields very distinguishable improvement of the critical temperature, as well.     

Extremity of the disordered phase in the Ising model on the Bethe lattice

We prove that the disordered Gibbs distribution in the ferromagnetic Ising model on the Bethe lattice is extreme forTT _c ^SG , whereT _c ^SG is the critical temperature of the spin glass model on the Bethe lattice, and it is not extreme forT _c ^SG .     

Bethe lattice representation for sandpiles

Avalanches in sandpiles are represented by a process of percolation in a Bethe lattice with a feedback mechanism. The results indicate that the frequency spectrum and probability distribution of avalanches provide a better resemblance to the experimental results than other models using cellular automata simulations. Apparent discrepancies between experiments performed by different authors are reconciled. Critical behavior is expressed here by the critical properties of percolation phenomena.     

Kinetic antiferromagnetism in the triangular lattice

We show that the motion of a single hole in the infinite-U Hubbard model with frustrated hopping leads to weak metallic antiferromagnetism of kinetic origin. An intimate relationship is demonstrated between the simplest versions of this problem in one and two dimensions, and two of the most subtle many body problems, namely, the Heisenberg Bethe ring in one dimension and the two-dimensional triangular lattice Heisenberg antiferromagnet.     

Thermodynamics of a model wih interacting annealed bond impurities on the Bethe lattice

A magnetic model is considered consisting of annealed, mutually repelling ferromagnetic bond impurities in an antiferromagnetic host lattice. Using recurrence relation techniques, the grand-canonical version of this model is solved on the three-coordinated Bethe lattice. A generic phase diagram is obtained containing, apart from the usual ferro- and antiferromagnetic regimes, two distinct incommensurate phases as well as a period-four modulated phase. Evidence is obtained that in one of the two incommensurate phases impurity pairing occurs.     

Discrete anchoring: Bulk-substrate coupling in lattice models

We formulate the anchoring problem for discrete-state lattice models. Anchoring is the selection of a bulk equilibrium state from a degenerate set of equivalent equilibrium states in semi-infinite samples in contact with a substrate, a phenomenon widely discussed in the context of liquid crystalline displays. As a concrete example we consider this problem for the three-state Potts model employing two different approximations, viz., a layered mean-field approximation and a Bethe lattice approach. The anchoring behaviour of the model is shown to be completely determined by the symmetry properties of the Hamiltonian.