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Using the method of recursion relations an exact solution of classical Ising models with arbitrary value of spin on the Bethe lattice with arbitrary coordination number is presented. Expressions for the spontaneous magnetization, for the magnetic moments of arbitrary orders, for the susceptibility, for the free energy, and for the specific heat are found as functions of quantities which are determined by the recursion relations. The behavior of the spontaneous magnetization for the Ising model on the Bethe lattice is investigated for systems with spin values up to s = 5 for various coordination numbers and the corresponding critical temperatures are determined. An approximate formula for determining the positions of the critical temperatures for arbitrary high values of the spin variable is found and discussed. It is shown that this formula allows one to determine the full structure of the critical temperatures with very high precision. [ABSTRACT FROM AUTHOR]

A lattice-based analysis of the percolation threshold for randomly distributed cylindrical particles is generalized to consider arbitrary joint distributions over the radii and lengths of the rods. Effects due to the finite hard core diameter of the particles are accounted for. An analogy to site percolation on a modified Bethe lattice is exploited to yield a result for the percolation threshold that is equivalent to one that has been obtained from integral equation methods in the limit of large aspect ratios for the rods. [ABSTRACT FROM AUTHOR]

Abstract: The effects of crystal field which is randomly turned on with probability p or turned off with probability 1−p on the sites of the Bethe lattice are investigated to study the critical behavior of the spin-3/2 Blume–Capel model. The order-parameters are obtained in terms of the recursion relations and then the possible phase diagrams are calculated at finite temperatures for given system parameters p, and coordination numbers z=3, 4, 5 and 6. It was found that the phase diagrams change drastically for given probability and randomness. The first-order phase transitions are only found for higher p values and the lines of which get shorter and eventually disappear as p decreases. The critical lines do not present any qualitative differences with z, but they are seen at higher temperatures for higher z. [Copyright &y& Elsevier]

Abstract: The stationary states of the kinetic spin-1 Blume–Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field () and crystal field () interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures () for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the and planes for the coordination number . In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed. [Copyright &y& Elsevier]

Abstract: The dependence of the phase diagrams on the random crystal field (RCF) is investigated for the spin-1 Blume–Capel (BC) model on the Bethe lattice. The calculations are carried out in terms of the recursion relations for the coordination number which corresponds to the square lattice. The model presents tricritical points which are observed at lower negative crystal fields and higher temperatures for higher probabilities and which vanish at lower ’s. The effect of randomness is illustrated for and shown that it changes the phase diagrams drastically from random to non-random systems. The reentrant behavior is also observed for appropriate values. [Copyright &y& Elsevier]

Abstract: The AB p C 1−p type of mixed ferromagnetic–ferrimagnetic ternary-alloy with A (spin-3/2), B (spin-1) and C (spin-5/2) ions was studied on the Bethe lattice with the odd numbered shells containing only A ions, while the even numbered shells either containing B or C ions randomly. The phase diagrams were obtained on the and (p, kT c /J AB ) planes for given values of p and R, respectively, with the coordination numbers z=3, 4, 5 and 6. The explicit dependence of the phase diagrams on z and each shell of the Bethe lattice having only one type of ion lead to some differences when compared with the previous works. The model presents one or two compensation temperatures for appropriate values of the system parameters. [ABSTRACT FROM AUTHOR]

Abstract: The ground-state and finite-temperature behavior of the mixed spin-1 and spin-1/2 Ising–Heisenberg model on the diamond-like decorated Bethe lattice is investigated within the framework of two rigorous methods: the decoration-iteration transformation and exact recursion relations. The model under consideration describes a hybrid classical-quantum system consisting of the Ising and Heisenberg spins, which interact among themselves either through the Ising or XXZ Heisenberg nearest-neighbor interaction. Both sublattice magnetizations of the Ising and Heisenberg spins are exactly calculated with the aim to examine phase diagrams, thermal variations of the total and sublattice magnetizations. The finite-temperature phase diagrams form continuous (second-order) phase transition lines only, which exhibit a small reentrant region if the diamond-like decorated Bethe lattice with a sufficiently high coordination number is considered. [ABSTRACT FROM AUTHOR]

Abstract: The Ising model for the spin- sandwiched trilayer Bethe lattices is investigated in terms of the recursion relations with either ferromagnetic (FM) or antiferromagnetic (AFM) type of bilinear interactions between the nearest-neighbor (NN) spins. In addition to the ground-state (GS) phase diagrams, the thermal variations of the order-parameters, spin–spin correlation functions and free energies are studied; therefore, different topological temperature-dependent phase diagrams are obtained. It was found that the system exhibits second- and first-order phase transitions. The bicritical, critical end and multicritical points, and the stable and unstable tricritical points are observed in addition to the reentrant behavior for the coordination numbers , 4 and 6. [Copyright &y& Elsevier]

We consider the increase in the number of trees as their size increases in the Eden growth model on simple and face-centered hypercubic lattices in different space dimensions. We propose a first-order partial differential equation for the tree generating function, which allows relating the exponent at the critical point of this function to the perimeter of the most probable tree. We estimate tree perimeters for the lattices considered. The theoretical values of the exponents agree well with the values previously obtained by computer modeling. We thus explain the closeness of the dimension dependences of the exponents of the simple and face-centered lattices and their difference from the results in the Bethe lattice approximation. [ABSTRACT FROM AUTHOR]

Abstract: The sandwiched trilayer of Bethe lattices in the form of the spins with spin-(1/2,1,1/2) Ising model is studied in terms of the recursion relations with either ferromagnetic or antiferromagnetic type bilinear interactions between the nearest-neighbor (NN) spins. The ground-state (GS) phase diagrams are obtained and it was found that the model presents six different GS phase configurations. In order to obtain the phase diagrams, the thermal variations of the order-parameter, spin–spin correlation functions and free energy are analyzed and different topological phase diagrams are obtained. It was found that the system exhibits different critical behaviors such as, second- and first-order phase transitions, tricritical and bicritical points for the values of the coordination numbers q=3,4 and 6. [ABSTRACT FROM AUTHOR]

Local vibrational density of states for disordered graphene has been calculated via Green’s functions method. Disordered material has been modeled with Bethe lattice. Density of states does not include particularities specific for ideal graphene. [ABSTRACT FROM AUTHOR]

Abstract: The spin-1 Ising model is studied for the case of antiferromagnetic (AFM)/AFM interactions on the bilayer Bethe lattice by using the pairwise approach for several values of the coordination numbers , 4 and 6 when the layers are linked with the external magnetic fields. The ground state (GS) phase diagrams, thermal variations of the order-parameters and the response functions are studied in detail to obtain the phase diagrams of the model. It was found that the system gives only one Néel temperature, , for . Two ''s and first-order phase transition temperature, , thus tricritical point, are found and the existence of two ''s leads to the reentrant behavior when and 6 for some given system parameters. [Copyright &y& Elsevier]

Abstract: We study the interplay between antiferromagnetism and the paramagnetic metal–insulator-transition (PMIT) on a Bethe lattice with nearest and next-nearest neighbor hopping. For the PMIT outgrows the antiferromagnetic phase and shows a scenario similar to V2O3. We also look at other ordered phases for the doped system, where we find a ferromagnetic phase. [Copyright &y& Elsevier]

The mixed spin-1/2 and spin-1 Ising model on the Bethe lattice with both uniaxial as well as biaxial single-ion anisotropy terms is solved exactly by combining star-triangle and triangle-star mapping transformations with exact recursion relations. Magnetic properties (magnetization, phase diagrams, and compensation phenomenon) are investigated in detail. Particular attention is focused on the effect of uniaxial and biaxial single-ion anisotropies that basically influence the magnetic behavior of the spin-1 atoms. [ABSTRACT FROM AUTHOR]

The distinct hysteresis loops (HLs) of ferromagnetic/antiferromagnetic (FM/AFM) two-layer Bethe lattice with the Ising spins of the top layer having only FM interactions and the bottom ones having only AFM interactions with the interlayer coupling is either FM or AFM type are studied by using a pairwise approach. The sublattice magnetizations are studied by increasing and decreasing the external magnetic field (H) to obtain the HLs. The shapes of the HLs are strongly dependent on the competitions among the system parameters and on the phase configurations. The HLs are formed only when the AFM-type interactions are involved. The small loops of hysteresis are also formed because of the reentrant behavior in the FM region. [ABSTRACT FROM AUTHOR]

Abstract: The magnetic properties of the ternary system ABC consisting of spins , S=1, and are investigated on the Bethe lattice by using the exact recursion relations. We consider both ferromagnetic and antiferromagnetic exchange interactions. The exact expressions for magnetizations and magnetic susceptibilities are found, and thermal behaviors of magnetizations and susceptibilities are studied. We construct the phase diagrams and find that the system exhibits one, two or even three compensation temperatures depending on the values of the interaction parameters in the Hamiltonian. Moreover, the system undergoes a second-order phase transition for the coordination number q⩽3 and a second- and first-order phase transitions for q>3; hence the system gives a tricritical point. The system also exhibits the reentrant behaviors. [Copyright &y& Elsevier]

Abstract: Two-layer Bethe lattice with the Ising spins of the top layer having only ferromagnetic (FM) interactions and the bottom layer having only antiferromagnetic (AFM) interactions are allowed to interact with the interlayer interaction of either FM or AFM type. The model is studied by using the exact recursion relations in a pairwise approach for given coordination numbers , 4 and 6 with equal external magnetic fields acting on the layers. The phase diagrams of the model are obtained on different planes for given system parameters by studying the ground state (GS) phase diagrams and the thermal variations of the order-parameters and the response functions, i.e. the susceptibility and the specific heat, in detail. The model presents second- and first-order phase transitions, and where their lines are combined is the tricritical point. The critical end points also exist. The reentrant behavior is also seen when the model presents two Néel temperatures. [Copyright &y& Elsevier]

Abstract: We have found an exact phase diagram of the Potts model with competing nearest neighbor and next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the multicritical point . We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by zero average magnetization lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting. [Copyright &y& Elsevier]

When gas molecules bind to a surface they may do so in such a way that the adsorption of one molecule inhibits the arrival of others. Two models which have frequently been studied are the "dimer model" and the "blocking model", and rather complete solutions for these are known on the Bethe lattice. In this letter, the probability that a connected set of sites remains unoccupied is calculated. [ABSTRACT FROM AUTHOR]

The bilayer spin-2 Ising model on the Bethe lattice is investigated by taking into account the intralayer coupling constants of the two layers J1 and J2, interlayer coupling constant between the layers J3 and crystal field interaction Δ by using the exact recursion equations in a pairwise approach. The ground state (GS) phase diagrams of the model are obtained on the (J2/|J1|, J3/q|J1|) planes for given Δ values and on the (Δ/qJ, J3/qJ) plane when J1 = J2 = J, and thus 33 distinct GS configurations are found. The temperature-dependent phase diagrams are obtained for J1 > 0, J2 > 0, and for J3 > 0 or J3 < 0 on the (kT/J1, J3/J1) planes for given Δ/qJ1 and J2/J1 and on the (Δ/J, kT/J) plane for given J3/J when J1 = J2 = J for the coordination number q = 3. It was found that the system exhibits both first- and second-order phase transitions and tricritical points. The paramagnetic phases are also classified by studying the thermal variations of the quadrupolar moments. [ABSTRACT FROM AUTHOR]

A spin-2 system consisting of two layers of Bethe lattices each with a branching ratio of q Ising spins was analyzed by the use of the exact recursion relations in a pairwise approach. The upper layer interacting with nearest-neighbor (NN) bilinear interaction J1 is laid over the top of the lower layer interacting with bilinear NN interaction J2, and the two layers are tied together via the bilinear interaction between the vertically aligned adjacent NN spins denoted as J3. The study of the ground state phase diagrams on the (J2/|J3|, J1/|J3|) plane with J3>0 and J3<0 and on the (J2/J1, J3/q J1) plane with J1>0 has yielded five distinct ground state configurations. The temperature dependent phase diagrams are obtained for the case with intralayer coupling constants of the two layers with ferromagnetic type J1 and J2>0, and the interlayer coupling constant of the layers with either ferromagnetic J3>0 or antiferromagnetic type J3<0 on the (kT/J1, J3/J1) planes for given values of the J2/J1 for various values of the coordination numbers. As a result, we have found that the model presents both second- and first-order phase transitions, therefore, tricritical points. [ABSTRACT FROM AUTHOR]

Abstract: The totally antiferromagnetic Ising model is analyzed on a bilayer Bethe lattice in detail by studying the order-parameters, response functions, i.e. susceptibility and specific heat, and free energy by using the recursion relations in a pairwise approach. The ground state phase diagrams of the model are also obtained on the plane for given values of and on the plane for given . As a result, we have obtained the temperature-dependent phase diagrams for various values of the coordination number q on the and planes for given values of the rest of the system parameters. [Copyright &y& Elsevier]

Using recently developed methods we have calculated exactly the correlation length amplitude ratios for Ising model on the Bethe lattice and found that this ratio is independent of the coordination number of the lattice. [ABSTRACT FROM AUTHOR]

Abstract: Equations for the spin glass order in the Ising spin glass model on the Bethe-like lattices with and without small loops are studied. For each lattice, equations are obtained by using and not using the replica method. Within the replica symmetric approximation, equations obtained by the two ways are shown to be identical. To see the effects of the small loops and the replica symmetry breaking, a spin glass order parameter is investigated as a function of the connectivity of the lattices close to the transition temperature. Replica symmetry breaking is enhanced by the existence of small loops. [Copyright &y& Elsevier]

Abstract: The influence of increasing the coordination number q on the phase diagrams of the mixed spin-2 and spin- Blume–Capel Ising ferrimagnetic system with two crystal-field interactions, for spin-2 and for spin-, is investigated on the Bethe lattice by using the exact recursion equations. The phase diagrams are obtained for the case with equal crystal field strengths when , 4 and 6 on the () planes and for the case with unequal crystal field strengths on the () and () planes for various values of and , respectively, for and 6. Besides the second-order lines, the model presents three types of first-order phase transition lines; one of them ends on the compensation lines (), the other one ends at a critical end point () and last one terminates at a tricritical point () for the case with equal crystal field strengths. In this case we have also observed up to four compensation temperatures when . Besides the second-order, tricritical and compensation lines, the system presents two types of the first-order lines one of them ending at a tricritical point and the other one ending at an isolated point when unequal crystal field values are taken. [Copyright &y& Elsevier]

The interactive two-state model of cell membrane ion channels in an electric field is formulated on the Bethe lattice by means of the exact recursion relations. The probability of channel opening or maximum fractions of open potassium and sodium channels are obtained by solving a non-linear algebraic equation. Using known parameters for the conventional mean-field theory the model gives a good agreement with the experiment both at low and high trans-membrane potential values. For intermediate voltages, the numerical results imply that collective effects are introduced by trans-membrane voltage. [ABSTRACT FROM AUTHOR]

Abstract: A bilayer Ising model consisting of two Bethe lattices each with a branching ratio of Ising spins with one of the layers having only spin- and the other having only spin- is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. The problem was studied by using the exact recursion relations in a pairwise approach in terms of the intralayer bilinear interactions and of the upper and lower layers, respectively, and the interlayer bilinear interaction . After obtaining the ground state phase diagrams on the () plane with either or , the variations of the order-parameters and the free energy were analyzed to obtain the temperature dependent phase diagrams. They were calculated for only the ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers. It was found that the system presents both second- and first-order phase transitions, besides the isolated critical and triple points. The model also presents compensation temperatures when of spin- layer can compete with of spin- layer. [Copyright &y& Elsevier]

Abstract: The phase diagrams of the bilayer spin- Ising model on the Bethe lattice is analyzed by taking into account the intralayer coupling constants of the two layers , interlayer coupling constant between the layers and crystal field interaction (D) for the coordination number by using the exact recursion equations. First, the distinct ground state phase diagrams of the model are obtained on the plane with for given crystal field values. Then, the phase diagrams of the system are obtained on the planes for given values of the crystal field and on the honeycomb lattice, i.e., . It was found that the system presents both first- and second-order phase transitions, therefore, tricritical points, and splitting of the critical lines at negative values of the crystal field. The paramagnetic phases are also classified by studying the thermal variations of the quadrupolar moments. [Copyright &y& Elsevier]

In this paper, I will present an alternative approach to the Bethe or cactus lattice approximation, widely employed in the theory of cooperative phenomena. This approach relies on a variational free energy, which is equivalent to the Bethe free energy in that it has the same stationary points, but allows one to simplify analytical calculations, since it is a function of only single-site probability distributions, in the same way as an ordinary mean-field (Bragg-Williams) free energy. As an application, I shall discuss a derivation of closed-form equations for critical points in Ising-like models. Moreover, I will suggest a rule of thumb to choose the cactus lattice connectivity yielding the best approximation for the corresponding model defined on an ordinary lattice. [ABSTRACT FROM AUTHOR]

The critical and compensation temperatures of the bilayer Bethe lattices with one of the layers having only spin-1/2 atoms and the other having only spin-1 atoms placed symmetrically are studied by using exact recursion relations in a pairwise approach. The Hamiltonian of the model consist of the bilinear intralayer coupling constants of the two layers J 1 and J 2 for the interactions of the atoms in layers with spin-1/2 and spin-1, respectively, and the bilinear interlayer coupling constant J 3 between the adjacent atoms with spin-1/2 and spin-1 of the layers. After obtaining the ground state phase diagram with J 1 > 0, the variations of the order-parameters and the free energy are investigated to obtain the phase diagram of the model by considering only the ferromagnetic ordering of the layers, i.e. J 1 > 0 and J 2 > 0, and ferromagnetic or antiferromagnetic ordering of the adjacent spins of the layers, J 3 > 0 or J 3 < 0, respectively. It was found that the system presents both second- and first-order phase transitions and, tricritical points. The compensation temperatures was also observed for the appropriate values of the system parameters. [ABSTRACT FROM AUTHOR]

Abstract: The lattice spin model with Q-component discrete spin variables restricted to have orientations orthogonal to the faces of Q-dimensional hypercube is considered on the Bethe lattice, the recursive graph which contains no cycles. The partition function of the model with dipole–dipole and quadrupole–quadrupole interaction for arbitrary planar graph is presented in terms of double graph expansions. The latter is calculated exactly in case of trees. The system of two recurrent relations (RR) which allows to calculate all thermodynamic characteristics of the model is obtained. The correspondence between thermodynamic phases and different types of fixed points of the RR is established. Using the technique of simple iterations the plots of the zero field magnetization and quadrupolar moment are obtained. Analyzing the regions of stability of different types of fixed points of the system of recurrent relations the phase diagrams of the model are plotted. For the phase diagram of the model is found to have three tricritical points, whereas for there are one triple and one tricritical points. [Copyright &y& Elsevier]

Abstract: The critical and the compensation temperatures of the mixed spin- and spin-2 Ising ferrimagnetic system is examined on the Bethe lattice by using the exact recursion equations. First, the ground state phase diagram of the system at zero temperature is obtained on the (, ) plane. Then in order to investigate the influences of the sublattice crystal fields at higher temperatures, the phase diagrams on the () and () planes are-calculated for given values of for spin-2 and for spin-, respectively. Besides the second-order phase transition lines, two types of the first-order lines one ending at a tricritical point and the other at an isolated critical point are found. The model also exhibits up to four compensation temperatures for appropriate values of the system parameters. The calculations are carried out for the coordination number corresponding to the honeycomb lattice. [Copyright &y& Elsevier]

Abstract: The spin- Ising model on the bilayer Bethe lattice is studied in terms of the exact recursion relations with the intralayer bilinear interactions and of the two layers with ferromagnetic coupling and interlayer bilinear interaction between the layers with ferromagnetic or antiferromagnetic coupling. We first have obtained the ground state configurations of the model on the planes for and . Then the thermal behaviors of the order-parameters, the total and staggered magnetizations of the two layers and also the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers, are studied to obtain the phase diagrams of the model on the plane for given values of and the coordination number . As a result, a few types of the critical temperatures of the model are obtained. [Copyright &y& Elsevier]

Abstract: The mixed spin-2 and spin- Blume–Emery–Griffiths (BEG) Ising ferrimagnetic system is studied on the Bethe lattice using the exact recursion equations with the coordination number corresponding to the honeycomb lattice on real lattices. The influences of the crystal field and the biquadratic interactions are investigated by obtaining the phase diagrams on the () and () planes, respectively, with equal crystal field interactions for the sublattices. The model presents very rich critical behaviors, which includes the first- and second-order phase transitions, thus also the tricritical and critical end points are observed. We have also found that the model gives up to four compensation temperatures for appropriate values of the system parameters. [Copyright &y& Elsevier]

Abstract: The mixed spin-2 and spin- Blume–Capel Ising ferrimagnetic system, i.e. negative bilinear interaction of the nearest-neighbor spins of the sublattices, with two crystal–field interactions, for spin-2 and for spin-, is investigated on the Bethe lattice by using the exact recursion equations. The exact expressions for the order-parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. Then, the phase diagrams on the and planes are studied for various values of and , respectively, when the coordination number . It is observed that the system presents both second- and first-order phase transitions, therefore, tricritical points, besides the compensation temperatures for the sublattice magnetizations. In addition to this, we have also found that the sublattice quadrupolar moments also compensate each other meaning that their values are equal, whenever the sublattice magnetizations compensate, but at different temperatures, which is a type of compensation not mentioned in the literature as far as in our knowledge. [Copyright &y& Elsevier]

Abstract: The antiferromagnetic Blume–Emery–Griffiths model in an external magnetic field is studied by using the exact recursion relations on the Bethe lattice for arbitrary values of biquadratic and for negative values of bilinear interactions. We have studied the thermal variations of two-sublattice magnetizations belonging to spin-1 BEG model to obtain the phase diagrams on the plane. As a result, we have found that the system presents second- and first-order phase transitions, therefore, tricritical points for appropriate values of , and q. We have also found that the second-order phase transition lines exhibit reentrant phenomena in temperature, besides it also shows reentrant phenomena for the first-order phase lines in external magnetic field for and 6. [Copyright &y& Elsevier]

The Glauber dynamics of disordered spin models with multi-spin interactions on sparse random graphs (Bethe lattices) is investigated. Such models undergo a dynamical glass transition upon decreasing the temperature or increasing the degree of constrainedness. Our analysis is based upon a detailed study of large scale rearrangements which control the slow dynamics of the system close to the dynamical transition. Particular attention is devoted to the neighborhood of a zero temperature tricritical point. Both the approach and several key results are conjectured to be valid in a considerably more general context. [ABSTRACT FROM AUTHOR]

Abstract: In order to study the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume–Capel Ising ferrimagnetic system, we have used the exact recursion relations on the Bethe lattice. The system was studied for the coordination numbers with , 4, 5 and 6, and the obtained phase diagrams are illustrated on the plane for constant values of , the reduced crystal field of the sublattice with spin-5/2, and on the plane for constant values of , the reduced crystal field of the sublattice with spin-3/2, for only, since the cases corresponding to , 5 and 6 reproduce results similar to the case for . In addition we have also presented the phase diagram with equal strengths of the crystal fields for , 4, 5 and 6. Besides the second- and first-order phase transitions, the system also exhibits compensation temperatures for appropriate values of the crystal fields. In this mixed spin system while the second-order phase transition lines never cut the reduced crystal field axes as in the single spin type spin-3/2 and spin-5/2 Ising models separately, the first-order phase transition lines never connect to the second-order phase transition lines and they end at the critical points, therefore the system does not give any tricritical points. In addition to this, this mixed-spin model exhibits one or two compensation temperatures depending on the values of the crystal fields, as a result the compensation temperature lines show reentrant behavior. [Copyright &y& Elsevier]

Abstract: The low-temperature behavior of the hysteresis loop in the Ising model with crystal-field under uniform longitudinal magnetic field is studied by exact recursion relations on the Bethe lattice. The field dependences of the magnetization are examined in detail. The hysteresis curves are obtained for different temperatures (T): It is found that the hysteresis loops disappear in a certain temperature value. A number of characteristic behaviors for field variations are obtained especially for antiferromagnetic interaction. [Copyright &y& Elsevier]

Abstract: The mixed spin- and spin-S Ising ferrimagnetic systems with a crystal field are studied within the framework of the exact recursion relations on the Bethe lattice. The phase diagrams and thermal behaviors of magnetizations are investigated numerically for the different values of lattice coordination numbers. We find that the present system shows behaviors different from pure Ising systems. The results show that there is no compensation points for the system with half-integer values. [Copyright &y& Elsevier]

Abstract: The roughening temperature of an Ising model on a Cayley tree with six nearest neighbors is obtained, giving a strong evidence that the roughening temperature of a three-dimensional cubic lattice is very close to and not to . [Copyright &y& Elsevier]

We introduce the model of independent percolation on general graphs with emphasis on the Bethe lattice, for which we prove the existence of a phase transition with the precise calculation of the critical value, and derive the value of the critical exponents $\gamma$ and $\b$. [ABSTRACT FROM AUTHOR]

Abstract: The magnetic properties of a mixed Ising ferro and ferrimagnetic system on a Bethe lattice, in which the two interpenetrating sublattices have spins that can take two values, , alternated with spins that can take four values, , ±, are studied. On the Bethe lattice, we obtain closed expressions for sublattice and total magnetizations. The thermal behaviors of magnetizations are examined in detail. We find some interesting phenomena in these quantities, due to applied longitudinal magnetic field. [Copyright &y& Elsevier]

Abstract: The phase diagrams of the mixed spin-1 and spin-2 Blume–Capel Ising ferromagnetic system with two crystal-field interactions, for spin-1 and for spin-2, is studied on the Bethe lattice by using the exact recursion equations. The exact expressions of the order-parameters for the sublattices with spin-1 and spin-2, and the Curie temperature and the free energy of the system are obtained in terms of the recursion relations. The phase diagrams on the and planes are studied for selected values of and , respectively, for the coordination numbers , and 6. Besides the second-order phase transition lines, the lines of the first-order phase transitions terminating either at a tricritical point or at an isolated critical point are found. [Copyright &y& Elsevier]