Bond percolation and cluster formation on random lattices

1977 ◽  
Vol 55 (7-8) ◽  
pp. 635-645 ◽  
Author(s):  
H. J. Wintle ◽  
T. P. T. Williams
Keyword(s):  
Disordered Systems ◽  
Cayley Tree ◽  
Cluster Formation ◽  
Bond Number ◽  
Closed Loops ◽  
Coordination Numbers ◽  
Cayley Trees ◽  
Random Lattices ◽  
Bethe Lattices ◽  
Cyclic Systems

We have studied the formation of clusters and the distribution of bonds between sites on random lattices by Monte Carlo and analytic techniques for coordination numbers in the range 3 ≤ z ≤ 12. A comparison between Cayley trees (Bethe lattices) and systems in which closed loops are allowed (cyclic systems) indicates little difference in cluster formation but considerable differences in bond distribution between these two types of lattice. The results suggest that there is little difference between the percolation limits for the two types (at constant z), contrary to the existing results for disordered systems. This work points out a possible weakness in the analytic treatments of the Cayley tree due to the omission of correlation effects, and it also suggests that stochastic treatments of cyclic systems overestimate the critical bond number for percolation.

Magnetic properties of the mixed spin-5/2 and spin-3/2 Blume-Capel Ising system on the two-fold Cayley tree

Open Physics ◽  
2009 ◽  
Vol 7 (3) ◽  
Author(s):  
Rachidi Yessoufou ◽  
Saliou Amoussa ◽  
Felix Hontinfinde
Keyword(s):  
Magnetic Properties ◽  
Phase Transitions ◽  
Crystal Field ◽  
Cayley Tree ◽  
Second Order ◽  
Coordination Numbers ◽  
First Order ◽  
Mixed Spin ◽  
Second Order Transition ◽  
Zero Values

AbstractWe use exact recursion relations to study the magnetic properties of the half-integer mixed spin-5/2 and spin-3/2 Blume-Capel Ising ferromagnetic system on the two-fold Cayley tree that consists of two sublattices A and B. Two positive crystal-field interactions Δ1 and Δ2 are considered for the sublattice with spin-5/2 and spin-3/2 respectively. For different coordination numbers q of the Cayley tree sites, the phase diagrams of the model are presented with a special emphasis on the case q = 3, since other values of q reproduce similar results. First, the T = 0 phase diagram is illustrated in the (D A = Δ1/J,D B = Δ2/J) plane of reduced crystal-field interactions. This diagram shows triple points and coexistence lines between thermodynamically stable phases. Secondly, the thermal variation of the magnetization belonging to each sublattice for some coordination numbers q are investigated as well as the Helmoltz free energy of the system. First-order and second-order phase transitions are found. The second-order phase transitions become sharper and sharper when D A or D B increases. The first-order transitions only exist for some appropriate non-zero values of D A and/or D B. The corresponding transition lines never connect to the second-order transition lines. Thus, the non-existence of tricritical points remains one of the key features of the present model. The magnetic exponent β 0 of the model is estimated and found to be ¼ at small values of D A = D B = D and β 0 = ½ at large values of D. At intermediate values of D, there is a crossover region where the magnetic exponent displays interesting behaviours.

scholarly journals Cluster formation in disordered systems and nuclear fragmentation

1995 ◽  
Vol 592 (3) ◽  
pp. 385-412 ◽  
Author(s):  
B. Elattari ◽  
J. Richert ◽  
P. Wagner ◽  
Y.M. Zheng
Keyword(s):  
Disordered Systems ◽  
Cluster Formation ◽  
Nuclear Fragmentation

Cayley Tree Random Walk Dynamics

2000 ◽  
Vol 651 ◽  
Author(s):  
Dimitrios Katsoulis ◽  
Panos Argyrakis ◽  
Alexander Pimenov ◽  
Lexei Vitukhnovsky
Keyword(s):  
Random Walk ◽  
Cayley Tree ◽  
Infinite System ◽  
Mean Square Displacement ◽  
Finite Size ◽  
Finite Size Effects ◽  
Cayley Trees ◽  
Large Trees ◽  
The Mean ◽  
Diffusion Properties

AbstractWe investigate diffusion on newly synthesized molecules with dendrimer structures. We model these structures with geometrical Cayley trees. We focus on diffusion properties, such as the excursion distance, the mean square displacement of the diffusing particles, and the area probed, as given by the walk parameter SN, the number of the distinct sites visited, on different coordination number, z, and different generation order g of a dendrimer structure. We simulate the trapping kinetics curves for randomly distributed traps on these structures, and compare the finite and the infinite system cases, and also with the cases of regular dimensionality lattices. For small dendrimer structures, SN approaches the overall number of the dendrimer nodes, while for large trees it grows linearly with time. The average displacement R also grows linearly with time. We find that the random walk on Cayley trees, due to the nature ot these structures, is indeed a type of a “biased” walk. Finally we find that the finite-size effects are particularly important in these structures.

A Less Arbitrary Determination of Coordination Numbers in Disordered Systems

1986 ◽  
Vol 16 (1) ◽  
pp. 47-54 ◽  
Author(s):  
R. L. McGreevy ◽  
A. Baranyai ◽  
I. Ruff
Keyword(s):  
Disordered Systems ◽  
Coordination Numbers

scholarly journals Extremality of Disordered Phase of λ-Model on Cayley Trees

Algorithms ◽  
2022 ◽  
Vol 15 (1) ◽  
pp. 18
Author(s):  
Farrukh Mukhamedov
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Gibbs Measure ◽  
Cayley Tree ◽  
Communication Theory ◽  
Arbitrary Order ◽  
Reconstruction Problem ◽  
Free Boundary Conditions ◽  
Cayley Trees ◽  
Disordered Phase

In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.

scholarly journals On Topological Indices of Fractal and Cayley Tree Type Dendrimers

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Imran ◽  
Abdul Qudair Baig ◽  
Waqas Khalid
Keyword(s):  
Applied Sciences ◽  
Cayley Tree ◽  
Topological Indices ◽  
Topological Descriptors ◽  
Zagreb Index ◽  
Physiochemical Properties ◽  
Cayley Trees ◽  
Augmented Zagreb Index ◽  
Numerical Invariants ◽  
Atom Bond

The topological descriptors are the numerical invariants associated with a chemical graph and are helpful in predicting their bioactivity and physiochemical properties. These descriptors are studied and used in mathematical chemistry, medicines, and drugs designs and in other areas of applied sciences. In this paper, we study the two chemical trees, namely, the fractal tree and Cayley tree. We also compute their topological indices based on degree concept. These indices include atom bond connectivity index, geometric arithmetic index and their fourth and fifth versions, Sanskruti index, augmented Zagreb index, first and second Zagreb indices, and general Randic index for α={-1,1,1/2,-1/2}. Furthermore, we give closed analytical results of these indices for fractal trees and Cayley trees.

GIBBS MEASURES ON CAYLEY TREES: RESULTS AND OPEN PROBLEMS

2013 ◽  
Vol 25 (01) ◽  
pp. 1330001 ◽  
Author(s):  
UTKIR A. ROZIKOV
Keyword(s):  
Review Paper ◽  
Gibbs Measures ◽  
Modern Theory ◽  
Information Flows ◽  
Open Problems ◽  
Rigorous Results ◽  
Random Field Theory ◽  
Cayley Trees ◽  
Markov Random ◽  
Bethe Lattices

The purpose of this review paper is to present systematically all known results on Gibbs measures on Cayley trees (Bethe lattices). There are about 150 papers which contain mathematically rigorous results about Gibbs measures on Cayley trees. This review is mainly based on the recently published mathematical papers. The method used for the description of Gibbs measures on Cayley trees is the method of Markov random field theory and recurrent equations of this theory, but the modern theory of Gibbs measures on trees uses new tools such as group theory, information flows on trees, node-weighted random walks, contour methods on trees and nonlinear analysis. We discuss all the mentioned methods which were developed recently. Thus, the paper informs the reader about what is (mathematically) done in the theory of Gibbs measures on trees and about where the corresponding results were published. We only give proofs which were not published in literature. Moreover, we give several open problems.

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