13,858 research outputs found
The spectral dimension of random trees
We present a simple yet rigorous approach to the determination of the
spectral dimension of random trees, based on the study of the massless limit of
the Gaussian model on such trees. As a byproduct, we obtain evidence in favor
of a new scaling hypothesis for the Gaussian model on generic bounded graphs
and in favor of a previously conjectured exact relation between spectral and
connectivity dimensions on more general tree-like structures.Comment: 14 pages, 2 eps figures, revtex4. Revised version: changes in section
I
The statistical geometry of scale-free random trees
The properties of scale-free random trees are investigated using both
preconditioning on non-extinction and fixed size averages, in order to study
the thermodynamic limit. The scaling form of volume probability is found, the
connectivity dimensions are determined and compared with other exponents which
describe the growth. The (local) spectral dimension is also determined, through
the study of the massless limit of the Gaussian model on such trees.Comment: 21 pages, 2 figures, revtex4, minor changes (published version
Ising models on power-law random graphs
We study a ferromagnetic Ising model on random graphs with a power-law degree
distribution and compute the thermodynamic limit of the pressure when the mean
degree is finite (degree exponent ), for which the random graph has a
tree-like structure. For this, we adapt and simplify an analysis by Dembo and
Montanari, which assumes finite variance degrees (). We further
identify the thermodynamic limits of various physical quantities, such as the
magnetization and the internal energy
Maximal entropy random networks with given degree distribution
Using a maximum entropy principle to assign a statistical weight to any
graph, we introduce a model of random graphs with arbitrary degree distribution
in the framework of standard statistical mechanics. We compute the free energy
and the distribution of connected components. We determine the size of the
percolation cluster above the percolation threshold. The conditional degree
distribution on the percolation cluster is also given. We briefly present the
analogous discussion for oriented graphs, giving for example the percolation
criterion.Comment: 22 pages, LateX, no figur
Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe Lattices
We show that negative of the number of floppy modes behaves as a free energy
for both connectivity and rigidity percolation, and we illustrate this result
using Bethe lattices. The rigidity transition on Bethe lattices is found to be
first order at a bond concentration close to that predicted by Maxwell
constraint counting. We calculate the probability of a bond being on the
infinite cluster and also on the overconstrained part of the infinite cluster,
and show how a specific heat can be defined as the second derivative of the
free energy. We demonstrate that the Bethe lattice solution is equivalent to
that of the random bond model, where points are joined randomly (with equal
probability at all length scales) to have a given coordination, and then
subsequently bonds are randomly removed.Comment: RevTeX 11 pages + epsfig embedded figures. Submitted to Phys. Rev.
Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists
We review critically the concepts and the applications of Cayley Trees and
Bethe Lattices in statistical mechanics in a tentative effort to remove
widespread misuse of these simple, but yet important - and different - ideal
graphs. We illustrate, in particular, two rigorous techniques to deal with
Bethe Lattices, based respectively on self-similarity and on the Kolmogorov
consistency theorem, linking the latter with the Cavity and Belief Propagation
methods, more known to the physics community.Comment: 10 pages, 2 figure
Slow dynamics and rare-region effects in the contact process on weighted tree networks
We show that generic, slow dynamics can occur in the contact process on
complex networks with a tree-like structure and a superimposed weight pattern,
in the absence of additional (non-topological) sources of quenched disorder.
The slow dynamics is induced by rare-region effects occurring on correlated
subspaces of vertices connected by large weight edges, and manifests in the
form of a smeared phase transition. We conjecture that more sophisticated
network motifs could be able to induce Griffiths phases, as a consequence of
purely topological disorder.Comment: 12 pages, 10 figures, final version appeared in PR
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