674 research outputs found
Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees
We study the effect of boundary conditions on the relaxation time of the
Glauber dynamics for the hard-core model on the tree. The hard-core model is
defined on the set of independent sets weighted by a parameter ,
called the activity. The Glauber dynamics is the Markov chain that updates a
randomly chosen vertex in each step. On the infinite tree with branching factor
, the hard-core model can be equivalently defined as a broadcasting process
with a parameter which is the positive solution to
, and vertices are occupied with probability
when their parent is unoccupied. This broadcasting process
undergoes a phase transition between the so-called reconstruction and
non-reconstruction regions at . Reconstruction has
been of considerable interest recently since it appears to be intimately
connected to the efficiency of local algorithms on locally tree-like graphs,
such as sparse random graphs. In this paper we show that the relaxation time of
the Glauber dynamics on regular -ary trees of height and
vertices, undergoes a phase transition around the reconstruction threshold. In
particular, we construct a boundary condition for which the relaxation time
slows down at the reconstruction threshold. More precisely, for any , for with any boundary condition, the relaxation time is
and . In contrast, above the reconstruction
threshold we show that for every , for ,
the relaxation time on with any boundary condition is , and we construct a boundary condition where the relaxation time is
Glauber dynamics on trees:Boundary conditions and mixing time
We give the first comprehensive analysis of the effect of boundary conditions
on the mixing time of the Glauber dynamics in the so-called Bethe
approximation. Specifically, we show that spectral gap and the log-Sobolev
constant of the Glauber dynamics for the Ising model on an n-vertex regular
tree with plus-boundary are bounded below by a constant independent of n at all
temperatures and all external fields. This implies that the mixing time is
O(log n) (in contrast to the free boundary case, where it is not bounded by any
fixed polynomial at low temperatures). In addition, our methods yield simpler
proofs and stronger results for the spectral gap and log-Sobolev constant in
the regime where there are multiple phases but the mixing time is insensitive
to the boundary condition. Our techniques also apply to a much wider class of
models, including those with hard-core constraints like the antiferromagnetic
Potts model at zero temperature (proper colorings) and the hard--core lattice
gas (independent sets)
Phase transition for the mixing time of the Glauber dynamics for coloring regular trees
We prove that the mixing time of the Glauber dynamics for random k-colorings
of the complete tree with branching factor b undergoes a phase transition at
. Our main result shows nearly sharp bounds on the mixing
time of the dynamics on the complete tree with n vertices for
colors with constant C. For we prove the mixing time is
. On the other side, for the mixing time
experiences a slowing down; in particular, we prove it is
and . The critical point C=1
is interesting since it coincides (at least up to first order) with the
so-called reconstruction threshold which was recently established by Sly. The
reconstruction threshold has been of considerable interest recently since it
appears to have close connections to the efficiency of certain local
algorithms, and this work was inspired by our attempt to understand these
connections in this particular setting.Comment: Published in at http://dx.doi.org/10.1214/11-AAP833 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.Comment: To appear in SIAM J. Computin
Spatial Mixing of Coloring Random Graphs
We study the strong spatial mixing (decay of correlation) property of proper
-colorings of random graph with a fixed . The strong spatial
mixing of coloring and related models have been extensively studied on graphs
with bounded maximum degree. However, for typical classes of graphs with
bounded average degree, such as , an easy counterexample shows that
colorings do not exhibit strong spatial mixing with high probability.
Nevertheless, we show that for with and
sufficiently large , with high probability proper -colorings of
random graph exhibit strong spatial mixing with respect to an
arbitrarily fixed vertex. This is the first strong spatial mixing result for
colorings of graphs with unbounded maximum degree. Our analysis of strong
spatial mixing establishes a block-wise correlation decay instead of the
standard point-wise decay, which may be of interest by itself, especially for
graphs with unbounded degree
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