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The molecular approach of a spin model is constructed on the Bethe lattice (BL), and then it is examined in terms of exact recursion relations. Rather than assuming that each BL site is inhabited by a single spin, each site is occupied by two spin-1/2 atoms A and B, forming a molecule. Each molecule is considered to contain two spin-1/2 atoms, as well as q = 3 , 4 , or 6 nearest-neighbor molecules. In addition to the internal interactions between the atoms of each molecule, the molecules interact via their atoms in terms of bilinear interaction parameters J. Atoms of a molecule interact with J A i B i , while the molecules interact via their atoms in terms of J A i B i + 1 = J B i A i + 1 and J A i A i + 1 = J B i B i + 1 . After obtaining the magnetizations of each atom in the central molecule of the BL, the average magnetization of the molecule is determined. It is found that the model presents first-and second-order and random phase transitions. The model also displays tricritical, bicritical and end points, in addition to reentrant behavior for appropriate J values. [ABSTRACT FROM AUTHOR]

The triangular-type Ising nanowire is constructed on the Bethe lattice (BL) by using the core-shell structure consisting of spin-3/2 atoms as the core and spin-1/2 atoms as the triangular shell. Each triangular plaquette of spins forms a nanoparticle which is connected to upper and lower plaquettes symmetrically. The additions of the plaquettes continue indefinitely until the thermodynamic limit to construct the nanowire. The inter- and intra-bilinear interaction parameters (J) are assumed to be positive or negative to simulate the ferromagnetic (FM) or antiferromagnetic (AFM) interactions, respectively. The crystal field for spin-3/2 and external magnetic field for all sites are also included into the model. After obtaining the formulation of the model in terms of exact recursion relations (ERRs), the thermal variations of magnetizations are studied in detail to obtain the phase diagrams. It is found that the model leads to different types of FM and AFM regions with various forms of phase transitions. It is also interesting that the model presents random or oscillatory magnetization behavior regions for the appropriate values of our system parameters. [ABSTRACT FROM AUTHOR]

Replica symmetry breaking (RSB) for spin glasses predicts that the equilibrium configuration at two different magnetic fields are maximally decorrelated. We show that this theory presents quantitative predictions for this chaotic behavior under the application of a vanishing external magnetic field, in the crossover region where the field intensity scales proportionally to 1/√N, being N the system size. We show that RSB theory provides universal predictions for chaotic behavior: They depend only on the zero-field overlap probability function P(q) and are independent of other system features. In the infinite volume limit, each spin-glass sample is characterized by an infinite number of states that have a tree-like structure. We generate the corresponding probability distribution through efficient sampling using a representation based on the Bolthausen-Sznitman coalescent. Using solely P(q) as input we can analytically compute the statistics of the states in the region of vanishing magnetic field. In this way, we can compute the overlap probability distribution in the presence of a small vanishing field and the increase of chaoticity when increasing the field. To test our computations, we have simulated the Bethe lattice spin glass and the 4D Edwards-Anderson model, finding in both cases excellent agreement with the universal predictions. [ABSTRACT FROM AUTHOR]

The exact recursion relations are used to study the mixed half-integer spin-(3/2, 7/2) Blume-Capel Ising ferrimagnetic system on the Bethe lattice. Ground-state phase diagrams are computed in the (DA /q|J|, DB /q|J|) plane to reveal different possible ground states of the model. Using the thermal changes of the order-arameters, interesting temperature dependent phase diagrams are constructed in the (DA/|J|, kT/|J|), (DB/|J|, kT/|J|) planes as well as in the (D/|J|, kT/|J|) plane where D = DA = DB. It is revealed that the system exhibits first- and second-order phase transitions and compensation temperatures for specific model parameter values. Under the constraint of an external magnetic field, the model also produces multi-hysteresis behaviors as single, double and triple hysteresis cycles. Particularly, the impacts of the ferrimagnetic coupling J on the remanent magnetization and the coercitive fields for selected values of the other physical parameters of the system are pointed out. Our numerical results are qualitatively consistent with those reported in the literature.

The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al (2022 Linear Multilinear Algebr. 72 325–65) proposed a classical 'Dirac walk' diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such 'Dirac walks' on the Bethe lattice, a d -regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ( d s = 1 for d = 2 and d s = 3 for d ⩾ 3 ). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Phase diagrams of the mixed spin- and Blume–Capel model on the Bethe lattice in the distribution with probabilities and for and and the adjustment parameter are obtained on the and planes for given values of single-ion anisotropy by varying in the range and in the range . The phase diagrams are constructed by studying the thermal variations of the order parameters in terms of exact recursion relations, and the probability distribution is then implemented into the model. The model presents first- and second-order phase transitions in addition to the tricritical and bicritical points for the coordination number corresponding to the honeycomb lattice. [ABSTRACT FROM AUTHOR]

Focusing on the diffusion-localization transition, we theoretically analyzed a nonlinear gravity-type transport model defined on networks called regular ring lattices, which have an intermediate structure between the complete graph and the simple ring. Exact eigenvalues were derived around steady states, and the values of the transition points were evaluated for the control parameter characterizing the nonlinearity. We also analyzed the case of the Bethe lattice (or Cayley tree) and found that the transition point is 1/2, which is the lowest value ever reported. [ABSTRACT FROM AUTHOR]

We consider the existence of the topological entropy of shift spaces on a finitely generated semigroup whose Cayley graph is a tree. The considered semigroups include free groups. On the other hand, the notion of stem entropy is introduced. For shift spaces on a strict free semigroup, the stem entropy coincides with the topological entropy. We reveal a sufficient condition for the existence of the stem entropy of shift spaces on a semigroup. Furthermore, we demonstrate that the topological entropy exists in many cases and is identical to the stem entropy. [ABSTRACT FROM AUTHOR]

Populations of isogenic embryonic stem cells or clonal bacteria often exhibit extensive phenotypic heterogeneity that arises from intrinsic stochastic dynamics of cells. The phenotypic state of a cell can be transmitted epigenetically in cell division, leading to correlations in the states of cells related by descent. The extent of these correlations is determined by the rates of transitions between the phenotypic states. Therefore, a snapshot of the phenotypes of a collection of cells with known genealogical structure contains information on phenotypic dynamics. Here, we use a model of phenotypic dynamics on a genealogical tree to define an inference method that allows extraction of an approximate probabilistic description of the dynamics from observed phenotype correlations as a function of the degree of kinship. The approach is tested and validated on the example of Pyoverdine dynamics in Pseudomonas aeruginosa colonies. Interestingly, we find that correlations among pairs and triples of distant relatives have a simple but nontrivial structure indicating that observed phenotypic dynamics on the genealogical tree is approximately conformal--a symmetry characteristic of critical behavior in physical systems. The proposed inference method is sufficiently general to be applied in any system where lineage information is available.

The outcome of the random crystal field effects on the antiferromagnetic spin-1 Blume–Capel model and external magnetic field are examined on the Bethe Lattice in terms of exact recursion relations. It is assumed that the crystal field is either turned on or off randomly with probability p and 1 − p , respectively. The phase diagrams are constructed from the thermal analysis of the order parameters with the coordination number z = 3 which corresponds to honeycomb lattice. It is explored that the system goes both second- and first-order phase transitions, along with the reentrant behavior and a few critical points. The reentrant behavior is stronger for lower values of p and disappears as p gets closer to 1.0. The first-order lines are observed to be either linked to the tricritical points or decomposed. The critical end points and double critical points are also observed. [ABSTRACT FROM AUTHOR]

The external random magnetic field (H i) with three nodes, i.e. acting up and down along the z -axis and zero, effective on the spins in the Blume-Capel model is analyzed on the Bethe lattice in terms of the exact recursion relations. All the nodes are assumed to have the same probability, p = 1 / 3 , so that the model could give various kinds of phase transitions. As a mapping of the phase transitions, the phase diagrams are constructed on two different planes which present very rich and interesting phase diagrams. In addition to the second- and first-order phase transitions, a few critical points, reentrant and double reentrant behaviors are also observed. [ABSTRACT FROM AUTHOR]

A triple mixed-spin Ising system defined on the Bethe lattice is numerically investigated by means of exact recursion relations (ERRs) calculations. The lattice is constituted by three types of magnetic atoms A, B, C with spins 1 / 2 , 1 , 3 / 2 respectively arranged in the form ABCABC. The effects of bilinear exchange and crystal-field interactions as well as those of thermal fluctuations on the order parameters and phase diagrams are thoroughly studied and specified. First-order transitions and tricritical points are present for the coordination number q = 4 whereas at q = 3 they are absent. Global compensation phenomena are absent for the magnetic system. Instead, it is shown that it can only occur between the sublattice magnetizations B and C of the system. Several novel kinds of reentrance of the phase boundaries while varying the values of model parameters have been reported. [ABSTRACT FROM AUTHOR]

The random bond-dilution effects of bilinear interaction parameter Jij between the nearest-neighbor (NN) sites are taken into consideration for the spin-1 Blume-Emery-Griffiths (BEG) model on the Bethe lattice (BL) comprised of two interpenetrating equivalent sublattices A and B for given coordination number z in terms of exact recursion relations (ERR). A bimodal distribution for Jij is assumed which is either introduced with probability p or closed with 1 − p. It is assumed that the biquadratic exchange interaction parameter (K) is constant between the NN spins and the single-ion anisotropy parameter (D) is taken to be equivalent on the sublattices A and B. After the study of thermal changes of the order-parameters, the phase diagrams are calculated on possible planes spanned by our system parameters. It is found that the model presents both first- and second-order phase transitions. In addition to the well-known ferromagnetic (F), paramagnetic (P) and ferrimagnetic (FI) phases, the staggered quadrupolar (SQ) phase is also observed. The bicritical point (BCP) for all z and double BCP with z ≥ 4 are observed. The tetracritical point was also found for lower values of p with z ≥ 5. [ABSTRACT FROM AUTHOR]

We analyze in detail a behavior of the order parameter in the half-filled extended Falicov-Kimball model for small Coulomb interactions (both onsite U and intersite V). The parameter is defined as the difference of localized electron concentrationsin both sublattices of the Bethelattice(in thelimitoflarge coordination number). Using two methods, namely, the dynamic mean field theory and the Hartree-Fock approximation, we found the ranges ofU and V for which the anomalous temperature dependence of the order parameter, characterized by the sharp reduction near T ~ Tc/2, occurs (Tc is the temperature of the continuous order-disorder transition). In order to quantitatively describe this anomaly, we defined a function that measures the departure of the order parameter dependence from the standard mean-field S = 1/2 Ising-like curve. We determined the Udependent critical value VC ofV above which the anomaly disappears. Indicators of the anomalous behavior of the parameter dependence can be also observed in the temperature dependence of the specific heat. [ABSTRACT FROM AUTHOR]

In this work, we have examined the thermal variations of the order parameters to calculate the phase diagrams for the mixed spin-1/2 and 5/2 Blume-Capel (BC) model on the Bethe lattice (BL) for the bimodal distribution of the bilinear exchange interaction, ± J, and an adjusting parameter, α, between them. In addition, the values of J are distributed on the BL with probabilities p and 1 − p for J > 0 corresponding to the ferromagnetic (FM) phase and J < 0 for the antiferromagnetic (AFM) phase, respectively. The calculations are carried out in terms of the exact recursion relations (ERR) and then the probability distribution is implemented into the model. The phase diagrams are obtained on the (D, T) and (p, T) planes by varying the rest of the parameters for the coordination number q = 3 which corresponds to honeycomb lattice. We have found in addition to the first- and second-order phase transitions, a few critical points such as tricritical and bicritical points. [ABSTRACT FROM AUTHOR]

The critical behaviors of the antiferromagnetic spin-3/2 Ising model are studied by using the exact recursion relations (ERR) in the presence of a random crystal field on the Bethe lattice (BL). The sublattice order parameters, magnetizations, and quadrupolar moments are obtained in terms of ERRs under the effect of random crystal field which was either turned on or off with probability p and 1 − p, respectively, in a bimodal form. The nature of phase transitions, thus, the phase diagrams of the model are calculated on the (H, T) planes for the randomly changing crystal field values with probability p for the coordination numbers q = 3 and 4. It is found that the model exhibits both second- and first-order phase transitions in addition to some critical points, i.e., tricritical point (TCP), critical end point, double critical point, and isolated end point. We have also observed the reentrant behavior for some values of our system parameters. [ABSTRACT FROM AUTHOR]

The A, B and C atoms with spin-1/2, spin-3/2 and spin-5/2 are joined together sequentially on the Bethe lattice in the form of ABCABC ... to simulate a molecule as a triple mixed-spin system. The spins are assumed to be interacting with only their nearest-neighbors via bilinear exchange interaction parameter in addition to crystal and external magnetic fields. The order-parameters are obtained in terms of exact recursion relations, then from the study of their thermal variations, the phase diagrams are calculated on the possible planes of our system. It is found that the model gives only second-order phase transitions in addition to the compensation temperatures. [ABSTRACT FROM AUTHOR]

The Ising model with nearest- and next-nearest-neighbor (NNN) bilinear interactions is examined on the Bethe lattice (BL) in terms of exact recursion relations (ERR) when the external magnetic H is turned on. The thermal variation of the magnetization belonging to the central spin is investigated to calculate the possible phase diagrams of the model for given coordination numbers. Different phase regions, ferromagnetic (FM), antiferromagnetic (AFM) and paramagnetic (PM), are discovered and the phase lines in terms of first-order or second-order phase transitions are calculated. These lines are found to be order–disorder or order–order phase transition lines. It is also found that they combine at some special points or terminate at some end points for appropriate values of the model parameters. [ABSTRACT FROM AUTHOR]

We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schrödinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to a sharp decay at any time. We then adapt the real variable methods first introduced by Escauriaza, Kenig, Ponce and Vega to establish a general sharp result in the case of bounded potentials. [ABSTRACT FROM AUTHOR]

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least θ occupied neighbors, occupied sites remain occupied forever. It is known that, when b>θ≥2, the limiting density q=q(p) of occupied sites exhibits a jump at some p T=p T(b,θ)∈(0,1) from q T:=q(p T)p T. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p T+h with h>0 and show that, as h ↓0, the system lingers around the “critical” state for time order h −1/2 and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q∈(q T,1) converges, as h ↓0, to a well-defined measure.

The isothermal entropy change of spin-1 Blume-Capel (BC) is investigated on the Bethe lattice (BL) with the variations of coordination numbers (q), crystal field (D) and external magnetic field (H) in the vicinity of critical temperatures, i.e. about the second- and first-order phase transition temperatures. The calculation is carried on the BL in terms of exact recursion relations. It is found that the peaks of the isothermal entropy variation obtained for both magnetization and quadrupole moments are observed to be increasing with increasing values of H for given D and q, decreasing with the increasing q for fixed D and H and also decreasing with the increasing values of D for constant q and H. The peaks of the curves for the quadrupole moment is less sharper then the ones obtained for the magnetization. [ABSTRACT FROM AUTHOR]

for the Ising model with nonmagnetic dilution, we consider a method for constructing the "pseudochaotic" impurity distribution based on the condition that the position correlation of movable impurity atoms in neighboring sites vanishes. For the one-dimensional Ising model with nonmagnetic dilution, we find the exact solution and show that the pseudochaotic approximation method gives the exact value of the magnetic susceptibility for this model in a zero external field. We assume that the pseudochaotic impurity distribution is completely uncorrelated in the region of zero magnetization for any lattice. This assumption is based on calculating the correlation functions for the Ising model with nonmagnetic dilution on the Bethe lattice. We find the magnetic susceptibility for that model. [ABSTRACT FROM AUTHOR]

The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes. [ABSTRACT FROM AUTHOR]

Abstract The ferrimagnetic-ferromagnetic-ferrimagnetic quaternary alloy is constructed on the Bethe lattice in the compound form as AB p C q D r and formulated in terms of the exact recursion relations in the standard-random approach. The QA is designed on the BL by placing the A atoms (spin-1) only on the sites of odd shells and randomly placing B (spin-3/2), C (spin-5/2) or D (spin-2) atoms with probabilities, or concentrations, p , q and r , respectively, on the sites of even shells. The phase diagrams are obtained from the thermal analysis of the order-parameters. It is also found that the model yields only one compensation temperature. Highlights • The ferrimagnetic‐ferromagnetic‐ferrimagnetic quaternary alloy (QA) is constructed on the Bethe lattice. • The QA is taken in the compound form as ABpCqDr and formulated in terms of exact recursion relations. • The randomization is applied in the standard‐random approach. • The A, B, C and D atoms are taken as spin‐1, spin‐3/2, spin‐5/2 and spin‐2, respectively. • The phase diagrams are studied from the thermal analysis of the order‐parameters. [ABSTRACT FROM AUTHOR]

The coordination numbers z = 3, 4 and 6 corresponding to the honeycomb, square, and simple cubic lattices, respectively, are randomly taken with probabilities p; q; r between the nearest-neighbors of spin-1 atoms on the Bethe lattice to simulate amorphous materials. The exact recursion relations are employed to obtain the orderparameters with the implementation of random coordination numbers in the standard-random approach. The phase diagrams are obtained by varying the temperature for given values of crystal field, coordination numbers, and probabilities. It is found that the phase lines consist of either second- or first-order phase transitions and secondand first-order phase transitions combined at the tricritical points for appropriate values of our parameters. [ABSTRACT FROM AUTHOR]

The spin-1 Blume-Capel model is investigated under the random change of coordination numbers on the Bethe lattice (BL). Various coordination numbers are taken as couples and varied randomly on the shells of the BL. The randomization is realized in the standard-random approach and the formulation is obtained in terms of exact recursion relations. The phase diagrams are obtained by calculating the places of the second- and first-order phase transitions and tricritical points. [ABSTRACT FROM AUTHOR]

We obtain a solution for the Potts model on the Bethe lattice in an external magnetic field with movable nonmagnetic impurities. Using the method of "pseudochaotic" impurity distribution (correlations in the positions of the impurity atoms for the neighboring sides vanish), we obtain a system of equations defining the first-order phase transition curve on the "temperature-external field" plane. We find the dependence of the endpoint of the phase transition line on the concentration of magnetic atoms. [ABSTRACT FROM AUTHOR]

The mixed spin-1/2 and spin-3/2 Blume–Capel (BC) model is considered on the Bethe lattice (BL) with randomly changing coordination numbers (CN) and examined in terms of exact recursion relations. A couple of two different CNs are changed randomly on the shells of the BL in terms of a standard–random approach to obtain the phase diagrams on possible planes of the system parameters. It is found from the thermal analysis of the order-parameters that the model only gives the second-order phase transitions as in the regular mixed case. As the probability of having larger CN increases, the temperatures of the critical lines also increase as expected. [ABSTRACT FROM AUTHOR]

The quaternary alloy (QA) is simulated on the Bethe lattice (BL) in the form of ABpCqDr and its phase diagrams are calculated by using the exact recursion relations (ERR) for the coordination number z = 3. The QA is designed on the BL by placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2), C (spin-5/2) or D (spin-1) atoms with probabilities p, q and r, respectively, on the even shells. A compact form of formulation for the QA is obtained in the standard-random approach which can easily be reduced to ternary alloy (TA) and mixed-spin models by the appropriate values of the random variables p, q and r. The phase diagrams are calculated on the temperature and ratio of bilinear interaction parameter planes for given values of probabilities. [ABSTRACT FROM AUTHOR]

In this work, the ternary alloy (TA) of the form ABpC1−p with spin-32, spin-2 and spin-52, respectively, is studied on the Bethe lattice in terms of exact recursion relations in the standard random approach. The bilinear interaction parameter JAB is assumed to be ferromagnetic between the nearest-neighbor spins with spin-32 and spin-2, while JAC is taken to be antiferromagnetic between spin-32 and spin-52. The possible phase diagrams are obtained from the thermal analysis of the order parameters for the given coordination numbers z = 3,4,5 and 6. This analysis has also revealed that the model gives both second- and first-order phase transitions in addition to the compensation temperatures. [ABSTRACT FROM AUTHOR]

In this work, we have focused on the random crystal field effects on the phase diagrams of the mixed spin-1 and spin-5/2 Ising system obtained by utilizing the exact recursion relations (ERR) on the Bethe lattice (BL). The distribution function P ( D i ) = p δ [ D i − D ( 1 + α ) ] + ( 1 − p ) δ [ D i − D ( 1 − α ) ] is used to randomize the crystal field.The phase diagrams are found to exhibit second- and first-order phase transitions depending on the values of α , D and p . It is also observed that the model displays tricritical point, isolated point, critical end point and three compensation temperatures for suitable values of the system parameters. [ABSTRACT FROM AUTHOR]

The effect of single-ion anisotropy on the thermal properties of the ternary-alloy in the form of A B p C 1 − p is investigated on the Bethe lattice (BL) in terms of exact recursion relations. The simulation on the BL consists of placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2) or C (spin-5/2) atoms with concentrations p and 1 − p , respectively, on the even shells. The phase diagrams are calculated in possible planes spanned by the system parameters: temperature, single-ion anisotropy, concentration and ratio of the bilinear interaction parameters for z = 3 corresponding to the honeycomb lattice. It is found that the crystal field drives the system to the lowest possible state therefore reducing the temperatures of the critical lines in agreement with the literature. [ABSTRACT FROM AUTHOR]

The Bethe lattice approach is used to examine the effect of a random crystal field (RCF) on the mixed spin-1/2 and 5/2 Blume-Capel model. A bimodal form of RCF is considered which either turns the crystal field on or off for given probabilities p and 1 − p, respectively, for the sites with spin-5/2. The exact recursion relations are employed to obtain all the characteristics of the model for given coordination numbers q = 3,4,5and 6. The thermal variations of the order-parameters are studied and net magnetizations are classified according to the usual letter coding. The phase diagrams are obtained on the probability-temperature and crystal field-temperature planes. It is found that the model yields only one compensation temperature for appropriate values of system parameters. [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.