2,284 research outputs found
Inapproximability for Antiferromagnetic Spin Systems in the Tree Non-Uniqueness Region
A remarkable connection has been established for antiferromagnetic 2-spin
systems, including the Ising and hard-core models, showing that the
computational complexity of approximating the partition function for graphs
with maximum degree D undergoes a phase transition that coincides with the
statistical physics uniqueness/non-uniqueness phase transition on the infinite
D-regular tree. Despite this clear picture for 2-spin systems, there is little
known for multi-spin systems. We present the first analog of the above
inapproximability results for multi-spin systems.
The main difficulty in previous inapproximability results was analyzing the
behavior of the model on random D-regular bipartite graphs, which served as the
gadget in the reduction. To this end one needs to understand the moments of the
partition function. Our key contribution is connecting: (i) induced matrix
norms, (ii) maxima of the expectation of the partition function, and (iii)
attractive fixed points of the associated tree recursions (belief propagation).
The view through matrix norms allows a simple and generic analysis of the
second moment for any spin system on random D-regular bipartite graphs. This
yields concentration results for any spin system in which one can analyze the
maxima of the first moment. The connection to fixed points of the tree
recursions enables an analysis of the maxima of the first moment for specific
models of interest.
For k-colorings we prove that for even k, in the tree non-uniqueness region
(which corresponds to k<D) it is NP-hard, unless NP=RP, to approximate the
number of colorings for triangle-free D-regular graphs. Our proof extends to
the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic
model under a mild condition
Bipartite stable Poisson graphs on R
Let red and blue points be distributed on according to two
independent Poisson processes and and let each red
(blue) point independently be equipped with a random number of half-edges
according to a probability distribution (). We consider
translation-invariant bipartite random graphs with vertex classes defined by
the point sets of and , respectively, generated by a
scheme based on the Gale-Shapley stable marriage for perfectly matching the
half-edges. Our main result is that, when all vertices have degree 2 almost
surely, then the resulting graph does not contain an infinite component. The
two-color model is hence qualitatively different from the one-color model,
where Deijfen, Holroyd and Peres have given strong evidence that there is an
infinite component. We also present simulation results for other degree
distributions
Threshold graph limits and random threshold graphs
We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice set of
examples for the emerging theory of graph limits.Comment: 47 pages, 8 figure
Percolation in invariant Poisson graphs with i.i.d. degrees
Let each point of a homogeneous Poisson process in R^d independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution mu on the positive integers. We consider
translation-invariant schemes for perfectly matching the stubs to obtain a
simple graph with degree distribution mu. Leaving aside degenerate cases, we
prove that for any mu there exist schemes that give only finite components as
well as schemes that give infinite components. For a particular matching scheme
that is a natural extension of Gale-Shapley stable marriage, we give sufficient
conditions on mu for the absence and presence of infinite components
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
We prove that there exist bipartite Ramanujan graphs of every degree and
every number of vertices. The proof is based on analyzing the expected
characteristic polynomial of a union of random perfect matchings, and involves
three ingredients: (1) a formula for the expected characteristic polynomial of
the sum of a regular graph with a random permutation of another regular graph,
(2) a proof that this expected polynomial is real rooted and that the family of
polynomials considered in this sum is an interlacing family, and (3) strong
bounds on the roots of the expected characteristic polynomial of a union of
random perfect matchings, established using the framework of finite free
convolutions we recently introduced
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
Independence ratio and random eigenvectors in transitive graphs
A theorem of Hoffman gives an upper bound on the independence ratio of
regular graphs in terms of the minimum of the spectrum of the
adjacency matrix. To complement this result we use random eigenvectors to gain
lower bounds in the vertex-transitive case. For example, we prove that the
independence ratio of a -regular transitive graph is at least
The same bound holds for infinite transitive graphs: we
construct factor of i.i.d. independent sets for which the probability that any
given vertex is in the set is at least . We also show that the set of
the distributions of factor of i.i.d. processes is not closed w.r.t. the weak
topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …