scholarly journals First encounters on Bethe lattices and Cayley trees

2020 ◽  
pp. 105594
Author(s):  
Junhao Peng ◽  
Trifce Sandev ◽  
Ljupco Kocarev
Keyword(s):  
Cayley Trees ◽  
Bethe Lattices

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.

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.

scholarly journals Lee-Yang zeros of the antiferromagnetic Ising model

2021 ◽  
pp. 1-35
Author(s):  
FERENC BENCS ◽  
PJOTR BUYS ◽  
LORENZO GUERINI ◽  
HAN PETERS
Keyword(s):  
Ising Model ◽  
Partition Functions ◽  
Cayley Trees ◽  
Circular Arcs ◽  
Precise Characterization ◽  
Ferromagnetic Ising Model ◽  
Location Of Zeros ◽  
Antiferromagnetic Ising Model ◽  
Recursively Defined Trees ◽  
Degree Bound

Abstract We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

ABSENCE OF PHASE TRANSITIONS ON TREE STRUCTURES

1993 ◽  
Vol 07 (29n30) ◽  
pp. 1947-1950 ◽  
Author(s):  
RAFFAELLA BURIONI ◽  
DAVIDE CASSI
Keyword(s):  
Long Range ◽  
Nearest Neighbor ◽  
Linear Chain ◽  
Mean Field ◽  
Range Order ◽  
The Other ◽  
Long Range Order ◽  
Field Phase ◽  
Tree Structures ◽  
Bethe Lattices

We rigorously prove that the correlation functions of any statistical model having a compact transitive symmetry group and nearest-neighbor interactions on any tree structure are equal to the corresponding ones on a linear chain. The exponential decay of the latter implies the absence of long-range order on any tree. On the other hand, for trees with exponential growth such as Bethe lattices, one can show the existence of a particular kind of mean field phase transition without long-range order.

scholarly journals Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices

1999 ◽  
Vol 59 (2) ◽  
pp. 2084-2092 ◽  
Author(s):  
P. M. Duxbury ◽  
D. J. Jacobs ◽  
M. F. Thorpe ◽  
C. Moukarzel
Keyword(s):  
Free Energy ◽  
Bethe Lattices
Sign in / Sign up

Export Citation Format

Share Document