The boson gas on a Cayley tree

We analyze the free boson gas on a Cayley tree using two alternative methods. The spectrum of the lattice Laplacian on a finite tree is obtained using a direct iterative method for solving the associated characteristic equation and also using a random walk representation for the corresponding fermion lattice gas. The existence of the thermodynamic limit for the pressure of the boson lattice gas is proven and it is shown that the model exhibits boson condensation into the ground state. The random walk representation is also used to derive an expression for the Bethe approximation to the infinite-volume spectrum. This spectrum turns out to be continuous instead of a dense point spectrum, but there is still boson condensation in this approximation.     

A class of solvable reaction-diffusion processes on a Cayley tree

Considering the most general one-species reaction-diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e. °°→•°, °°→•• and °•→••, and in the second model, only the diffusion process •°→°• exists. For the first model, the probabilities P_l(m;t), of finding m particles on the lth shell of the Cayley tree, have been found exactly, and for the second model, the functions P_l(1;t) have been calculated. It has been shown that these are the only integrable models if one restricts consideration to the L+1-shell probabilities P(m₀,m₁,…,m_L;t).     

Emergence of fractal scale-free networks from stochastic evolution on the Cayley tree

An unexpected recognition of fractal topology in some real-world scale-free networks has evoked again an interest in the mechanisms stimulating their evolution. To explain this phenomenon a few models of a deterministic construction as well as a probabilistic growth controlled by a tunable parameter have been proposed so far. A quite different approach based on the fully stochastic evolution of the fractal scale-free networks presented in this Letter counterpoises these former ideas. It is argued that the diffusive evolution of the network on the Cayley tree shapes its fractality, self-similarity and the branching number criticality without any control parameter. The last attribute of the scale-free network is an intrinsic property of the skeleton, a special type of spanning tree which determines its fractality.     

Diffusion-limited aggregates grown on nonuniform substrates

In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, p$p$, from the percolation threshold, _pc$p_c$ up to the jamming coverage, _pj$p_j$. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of p$p$, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different k$k$ sizes. Correspondingly, the fractal dimension of the aggregates increases as p$p$ raises in the same range _pc≤p≤_pj$p_c\le p\le p_j$. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.     

Interacting walkers on the Cayley tree,and polymer statistics

We obtain the generating function for an ensemble of random walkers on the Cayley tree of coordination numberz. The pair interaction between walkers is taken into account. This forbids two walkers to occupy the same lattice point after an equal number of steps. Interacting polymer statistics results from this model if one associates time (or the number of steps) with an additional space coordinate. The limiting free energy appears in a form that corresponds to the phase transition of 3/2 order.     

Phase diagram of the Ising Model on a Cayley tree in the presence of competing interactions and magnetic field

We study the phase diagram for the Ising Model on a Cayley tree with competing nearest-neighbor interactionsJ ₁ and next-nearest-neighbor interactionsJ ₂ andJ ₃ in the presence of an external magnetic field. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular cases, previous works by Vannimenus and by Inawashiroet al. At vanishing temperature, the phase diagram is fully determined, for all values and signs ofJ ₂/J ₁ andJ ₃/J ₂; in particular, we verify that values ofJ ₃/J ₂ high enough favor the paramagnetic phase. At finite temperatures, several interesting features (evolution of reentrances, separation of the modulated region into two disconnected pieces, etc.) are exhibited for typical values ofJ ₂/J ₁ andJ ₃/J ₂.Partially supported by the Brazilian Agencies CNPq and FINEP.     

Behavior of the effective fields for a regular ising model on the Cayley tree

A regular Ising model with nearest-neighbor interactions ofJ and–J(J>0) on a Cayley tree of coordination number 3 is investigated for the behavior of effective fields in a uniform external field. The effective fields show periodic and also aperiodic structures in the temperature-field plane. At absolute zero temperature, the equations determining effective fields are reduced to a nonlinear, one-dimensional, iterative equation. Arithmetic furcations of period and a screening of the furcations are observed.     

Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions

One of the main problems of statistical physics is to describe all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this article is to investigate phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions.     

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.     

Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

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.     

Exactly solvable reaction diffusion models on a Cayley tree

The most general reaction-diffusion model on a Cayley tree with nearest-neighbor interactions is introduced, which can be solved exactly through the empty-interval method. The stationary solutions of such models, as well as their dynamics, are discussed. Concerning the dynamics, the spectrum of the evolution Hamiltonian is found and shown to be discrete, hence there is a finite relaxation time in the evolution of the system towards its stationary state.     

Metastability in the Potts model on the Cayley tree

Metastability in the ferromagneticp-state Potts model defined on the Cayley tree is discussed. It is shown that the sign of the boundary fieldH _s determines the order of the transition as well as the stability of the low-temperature phase. Lowering the temperature withH _s >0, a system withp<2 (p>2) will display a second (first)-order transition to a metastable (stable) phase. ForH _s >0 a second (first)-order transition to a metastable (stable) phase occurs ifp>2 (p<2). In this case the system also has a residual entropy which is negative forp<2.     

An exact formulation of the Blume-Emery-Griffiths model on a two-fold Cayley tree model

A two-fold Cayley tree graph with fully q-coordinated sites is constructed and the spin-1 Ising Blume-Emery-Griffiths model on the constructed graph is solved exactly using the exact recursion equations for the coordination number q = 3. The exact phase diagrams in (kT/J, K/J ) and (kT/J, D/J) planes are obtained for various values of constants D/J and K/J, respectively, and the tricritical behavior is found. It is observed that when the negative biquadratic exchange (K) and the positive crystal-field (D) interactions are large enough, the tricritical point disappears in the (kT/J, K/J) plane. On the other hand, the system always exhibits a tricritical behavior in the phase diagram of (kT/J, D/J) plane. Received 8 June 2001 and Received in final form 28 September 2001     

Phase diagram of the one-state potts model on the Cayley tree

The phase diagram of the one-state Potts model on the closed asymmetric Cayley tree with branching ratior=2 is obtained from the Bethe-Peierls map. The route to chaos, via the period doubling cascade, is obtained by considering the antiferromagnetic coupling limit. The connection of the Potts model with the percolation problem is shown by calculating the order parameter, its susceptibility, the internal energy, and the specific heat as well as their asymptotic behavior at the paramagnetic-ferromagnetic critical point. Due to the type of the lattice and to the polynomial character of the map, this is the simplest known example of a McKay-Berker-Kirkpatrick spin-glass.