47,708 research outputs found
Resolvent of Large Random Graphs
We analyze the convergence of the spectrum of large random graphs to the
spectrum of a limit infinite graph. We apply these results to graphs converging
locally to trees and derive a new formula for the Stieljes transform of the
spectral measure of such graphs. We illustrate our results on the uniform
regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We
sketch examples of application for weighted graphs, bipartite graphs and the
uniform spanning tree of n vertices.Comment: 21 pages, 1 figur
Ising models on locally tree-like graphs
We consider ferromagnetic Ising models on graphs that converge locally to
trees. Examples include random regular graphs with bounded degree and uniformly
random graphs with bounded average degree. We prove that the "cavity"
prediction for the limiting free energy per spin is correct for any positive
temperature and external field. Further, local marginals can be approximated by
iterating a set of mean field (cavity) equations. Both results are achieved by
proving the local convergence of the Boltzmann distribution on the original
graph to the Boltzmann distribution on the appropriate infinite random tree.Comment: Published in at http://dx.doi.org/10.1214/09-AAP627 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Attraction time for strongly reinforced walks
We consider a class of strongly edge-reinforced random walks, where the
corresponding reinforcement weight function is nondecreasing. It is known, from
Limic and Tarr\`{e}s [Ann. Probab. (2007), to appear], that the attracting edge
emerges with probability 1 whenever the underlying graph is locally bounded. We
study the asymptotic behavior of the tail distribution of the (random) time of
attraction. In particular, we obtain exact (up to a multiplicative constant)
asymptotics if the underlying graph has two edges. Next, we show some
extensions in the setting of finite graphs, and infinite graphs with bounded
degree. As a corollary, we obtain the fact that if the reinforcement weight has
the form , , then (universally over finite graphs) the
expected time to attraction is infinite if and only if
.Comment: Published in at http://dx.doi.org/10.1214/08-AAP564 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Potts Models on Feynman Diagrams
We investigate numerically and analytically Potts models on ``thin'' random
graphs -- generic Feynman diagrams, using the idea that such models may be
expressed as the N --> 1 limit of a matrix model. The thin random graphs in
this limit are locally tree-like, in distinction to the ``fat'' random graphs
that appear in the planar Feynman diagram limit, more familiar from discretized
models of two dimensional gravity.
The interest of the thin graphs is that they give mean field theory behaviour
for spin models living on them without infinite range interactions or the
boundary problems of genuine tree-like structures such as the Bethe lattice.
q-state Potts models display a first order transition in the mean field for
q>2, so the thin graph Potts models provide a useful test case for exploring
discontinuous transitions in mean field theories in which many quantities can
be calculated explicitly in the saddle point approximation.Comment: 10 pages, latex, + 6 postscript figure
On the structure of random graphs with constant -balls
We continue the study of the properties of graphs in which the ball of radius
around each vertex induces a graph isomorphic to the ball of radius in
some fixed vertex-transitive graph , for various choices of and .
This is a natural extension of the study of regular graphs. More precisely, if
is a vertex-transitive graph and , we say a graph is
{\em -locally } if the ball of radius around each vertex of
induces a graph isomorphic to the graph induced by the ball of radius
around any vertex of . We consider the following random graph model: for
each , we let be a graph chosen uniformly at
random from the set of all unlabelled, -vertex graphs that are -locally
. We investigate the properties possessed by the random graph with
high probability, for various natural choices of and .
We prove that if is a Cayley graph of a torsion-free group of polynomial
growth, and is sufficiently large depending on , then the random graph
has largest component of order at most with high
probability, and has at least automorphisms with high
probability, where depends upon alone. Both properties are in
stark contrast to random -regular graphs, which correspond to the case where
is the infinite -regular tree. We also show that, under the same
hypotheses, the number of unlabelled, -vertex graphs that are -locally
grows like a stretched exponential in , again in contrast with
-regular graphs. In the case where is the standard Cayley graph of
, we obtain a much more precise enumeration result, and more
precise results on the properties of the random graph . Our proofs
use a mixture of results and techniques from geometry, group theory and
combinatorics.Comment: Minor changes. 57 page
Invariant Percolation and Harmonic Dirichlet Functions
The main goal of this paper is to answer question 1.10 and settle conjecture
1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions
on a graph to those of the infinite clusters in the uniqueness phase of
Bernoulli percolation. We extend the result to more general invariant
percolations, including the Random-Cluster model. We prove the existence of the
nonuniqueness phase for the Bernoulli percolation (and make some progress for
Random-Cluster model) on unimodular transitive locally finite graphs admitting
nonconstant harmonic Dirichlet functions. This is done by using the device of
Betti numbers.Comment: to appear in Geometric And Functional Analysis (GAFA
AKLT Models with Quantum Spin Glass Ground States
We study AKLT models on locally tree-like lattices of fixed connectivity and
find that they exhibit a variety of ground states depending upon the spin,
coordination and global (graph) topology. We find a) quantum paramagnetic or
valence bond solid ground states, b) critical and ordered N\'eel states on
bipartite infinite Cayley trees and c) critical and ordered quantum vector spin
glass states on random graphs of fixed connectivity. We argue, in consonance
with a previous analysis, that all phases are characterized by gaps to local
excitations. The spin glass states we report arise from random long ranged
loops which frustrate N\'eel ordering despite the lack of randomness in the
coupling strengths.Comment: 10 pages, 1 figur
Uniqueness of Gibbs states of a quantum system on graphs
Gibbs states of an infinite system of interacting quantum particles are
considered. Each particle moves on a compact Riemannian manifold and is
attached to a vertex of a graph (one particle per vertex). Two kinds of graphs
are studied: (a) a general graph with locally finite degree; (b) a graph with
globally bounded degree. In case (a), the uniqueness of Gibbs states is shown
under the condition that the interaction potentials are uniformly bounded by a
sufficiently small constant. In case (b), the interaction potentials are
random. In this case, under a certain condition imposed on the probability
distribution of these potentials the almost sure uniqueness of Gibbs states has
been shown.Comment: 9 page
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