746 research outputs found
The topological strong spatial mixing property and new conditions for pressure approximation
In the context of stationary nearest-neighbour Gibbs measures
satisfying strong spatial mixing, we present a new combinatorial
condition (the topological strong spatial mixing property (TSSM)) on the
support of sufficient for having an efficient approximation algorithm for
topological pressure. We establish many useful properties of TSSM for studying
strong spatial mixing on systems with hard constraints. We also show that TSSM
is, in fact, necessary for strong spatial mixing to hold at high rate. Part of
this work is an extension of results obtained by D. Gamarnik and D. Katz
(2009), and B. Marcus and R. Pavlov (2013), who gave a special representation
of topological pressure in terms of conditional probabilities.Comment: 40 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1309.1873 by other author
Representation and poly-time approximation for pressure of lattice models in the non-uniqueness region
We develop a new pressure representation theorem for nearest-neighbour Gibbs
interactions and apply this to obtain the existence of efficient algorithms for
approximating the pressure in the -dimensional ferromagnetic Potts,
multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply
to every inverse temperature but the critical. For Widom-Rowlinson and
hard-core, they apply to certain subsets of both the subcritical and
supercritical regions. The main novelty of our work is in the latter.Comment: 37 pages, 2 figure
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)
Rapid Mixing of Global Markov Chains via Spectral Independence: The Unbounded Degree Case
We consider spin systems on general n-vertex graphs of unbounded degree and explore the effects of spectral independence on the rate of convergence to equilibrium of global Markov chains. Spectral independence is a novel way of quantifying the decay of correlations in spin system models, which has significantly advanced the study of Markov chains for spin systems. We prove that whenever spectral independence holds, the popular Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on graphs of maximum degree ?, where ? is allowed to grow with n, converges in O((? log n)^c) steps where c > 0 is a constant independent of ? and n. We also show a similar mixing time bound for the block dynamics of general spin systems, again assuming that spectral independence holds. Finally, for monotone spin systems such as the Ising model and the hardcore model on bipartite graphs, we show that spectral independence implies that the mixing time of the systematic scan dynamics is O(?^c log n) for a constant c > 0 independent of ? and n. Systematic scan dynamics are widely popular but are notoriously difficult to analyze. This result implies optimal O(log n) mixing time bounds for any systematic scan dynamics of the ferromagnetic Ising model on general graphs up to the tree uniqueness threshold. Our main technical contribution is an improved factorization of the entropy functional: this is the common starting point for all our proofs. Specifically, we establish the so-called k-partite factorization of entropy with a constant that depends polynomially on the maximum degree of the graph
Consistent estimation of the basic neighborhood of Markov random fields
For Markov random fields on with finite state space, we
address the statistical estimation of the basic neighborhood, the smallest
region that determines the conditional distribution at a site on the condition
that the values at all other sites are given. A modification of the Bayesian
Information Criterion, replacing likelihood by pseudo-likelihood, is proved to
provide strongly consistent estimation from observing a realization of the
field on increasing finite regions: the estimated basic neighborhood equals the
true one eventually almost surely, not assuming any prior bound on the size of
the latter. Stationarity of the Markov field is not required, and phase
transition does not affect the results.Comment: Published at http://dx.doi.org/10.1214/009053605000000912 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Diffusion of energy in chains of oscillators with bulk noise
These notes are based on a mini-course given during the conference Particle
systems and PDE's - II which held at the Center of Mathematics of the
University of Minho in December 2013. We discuss the problem of normal and
anomalous diffusion of energy in systems of coupled oscillators perturbed by a
stochastic noise conserving energy
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