12,540 research outputs found
Rapid Mixing of Gibbs Sampling on Graphs that are Sparse on Average
In this work we show that for every and the Ising model defined
on , there exists a , such that for all with probability going to 1 as , the mixing time of the
dynamics on is polynomial in . Our results are the first
polynomial time mixing results proven for a natural model on for where the parameters of the model do not depend on . They also provide
a rare example where one can prove a polynomial time mixing of Gibbs sampler in
a situation where the actual mixing time is slower than n \polylog(n). Our
proof exploits in novel ways the local treelike structure of Erd\H{o}s-R\'enyi
random graphs, comparison and block dynamics arguments and a recent result of
Weitz.
Our results extend to much more general families of graphs which are sparse
in some average sense and to much more general interactions. In particular,
they apply to any graph for which every vertex of the graph has a
neighborhood of radius in which the induced sub-graph is a
tree union at most edges and where for each simple path in
the sum of the vertex degrees along the path is . Moreover, our
result apply also in the case of arbitrary external fields and provide the
first FPRAS for sampling the Ising distribution in this case. We finally
present a non Markov Chain algorithm for sampling the distribution which is
effective for a wider range of parameters. In particular, for it
applies for all external fields and , where is the critical point for decay of correlation for the Ising model on
.Comment: Corrected proof of Lemma 2.
Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree
We continue our study of the full set of translation-invariant splitting
Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for
the -state Potts model on a Cayley tree. In our previous work \cite{KRK} we
gave a full description of the TISGMs, and showed in particular that at
sufficiently low temperatures their number is .
In this paper we find some regions for the temperature parameter ensuring
that a given TISGM is (non-)extreme in the set of all Gibbs measures.
In particular we show the existence of a temperature interval for which there
are at least extremal TISGMs.
For the Cayley tree of order two we give explicit formulae and some numerical
values.Comment: 44 pages. To appear in Random Structures and Algorithm
Generalisation of the Hammersley-Clifford Theorem on Bipartite Graphs
The Hammersley-Clifford theorem states that if the support of a Markov random
field has a safe symbol then it is a Gibbs state with some nearest neighbour
interaction. In this paper we generalise the theorem with an added condition
that the underlying graph is bipartite. Taking inspiration from "Gibbs Measures
and Dismantlable Graphs" by Brightwell and Winkler we introduce a notion of
folding for configuration spaces called strong config-folding proving that if
all Markov random fields supported on are Gibbs with some nearest neighbour
interaction so are Markov random fields supported on the 'strong config-folds'
and 'strong config-unfolds' of .Comment: 27 pages, 7 figures; Typos corrected and some notation has been
change
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