35 research outputs found
Factor models on locally tree-like graphs
We consider homogeneous factor models on uniformly sparse graph sequences
converging locally to a (unimodular) random tree , and study the existence
of the free energy density , the limit of the log-partition function
divided by the number of vertices as tends to infinity. We provide a
new interpolation scheme and use it to prove existence of, and to explicitly
compute, the quantity subject to uniqueness of a relevant Gibbs measure
for the factor model on . By way of example we compute for the
independent set (or hard-core) model at low fugacity, for the ferromagnetic
Ising model at all parameter values, and for the ferromagnetic Potts model with
both weak enough and strong enough interactions. Even beyond uniqueness regimes
our interpolation provides useful explicit bounds on . In the regimes in
which we establish existence of the limit, we show that it coincides with the
Bethe free energy functional evaluated at a suitable fixed point of the belief
propagation (Bethe) recursions on . In the special case that has a
Galton-Watson law, this formula coincides with the nonrigorous "Bethe
prediction" obtained by statistical physicists using the "replica" or "cavity"
methods. Thus our work is a rigorous generalization of these heuristic
calculations to the broader class of sparse graph sequences converging locally
to trees. We also provide a variational characterization for the Bethe
prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotics of the partition function of Ising model on inhomogeneous random graphs
For a finite random graph, we defined a simple model of statistical
mechanics. We obtain an annealed asymptotic result for the random partition
function for this model on finite random graphs as n; the size of the graph is
very large. To obtain this result, we define the empirical bond distribution,
which enumerates the number of bonds between a given couple of spins, and
empirical spin distribution, which enumerates the number of sites having a
given spin on the spinned random graphs. For these empirical distributions we
extend the large deviation principle(LDP) to cover random graphs with
continuous colour laws. Applying Varandhan Lemma and this LDP to the
Hamiltonian of the Ising model defined on Erdos-Renyi graphs, expressed as a
function of the empirical distributions, we obtain our annealed asymptotic
result.Comment: 14 page
Belief Propagation on replica symmetric random factor graph models
According to physics predictions, the free energy of random factor graph
models that satisfy a certain "static replica symmetry" condition can be
calculated via the Belief Propagation message passing scheme [Krzakala et al.,
PNAS 2007]. Here we prove this conjecture for two general classes of random
factor graph models, namely Poisson random factor graphs and random regular
factor graphs. Specifically, we show that the messages constructed just as in
the case of acyclic factor graphs asymptotically satisfy the Belief Propagation
equations and that the free energy density is given by the Bethe free energy
formula
Short survey on stable polynomials, orientations and matchings
This is a short survey about the theory of stable polynomials and its
applications. It gives self-contained proofs of two theorems of Schrijver. One
of them asserts that for a --regular bipartite graph on vertices,
the number of perfect matchings, denoted by , satisfies
The other
theorem claims that for even the number of Eulerian orientations of a
--regular graph on vertices, denoted by , satisfies
To prove
these theorems we use the theory of stable polynomials, and give a common
generalization of the two theorems
Quenched central limit theorems for the Ising model on random graphs
The main goal of the paper is to prove central limit theorems for the
magnetization rescaled by for the Ising model on random graphs with
vertices. Both random quenched and averaged quenched measures are
considered. We work in the uniqueness regime or and
, where is the inverse temperature, is the critical
inverse temperature and is the external magnetic field. In the random
quenched setting our results apply to general tree-like random graphs (as
introduced by Dembo, Montanari and further studied by Dommers and the first and
third author) and our proof follows that of Ellis in . For the
averaged quenched setting, we specialize to two particular random graph models,
namely the 2-regular configuration model and the configuration model with
degrees 1 and 2. In these cases our proofs are based on explicit computations
relying on the solution of the one dimensional Ising models.Comment: 37 page