353 research outputs found
Thermodynamics of the Binary Symmetric Channel
We study a hidden Markov process which is the result of a transmission of the
binary symmetric Markov source over the memoryless binary symmetric channel.
This process has been studied extensively in Information Theory and is often
used as a benchmark case for the so-called denoising algorithms. Exploiting the
link between this process and the 1D Random Field Ising Model (RFIM), we are
able to identify the Gibbs potential of the resulting Hidden Markov process.
Moreover, we obtain a stronger bound on the memory decay rate. We conclude with
a discussion on implications of our results for the development of denoising
algorithms
Entropy and growth rate of periodic points of algebraic Z^d-actions
Expansive algebraic Z^d-actions corresponding to ideals are characterized by
the property that the complex variety of the ideal is disjoint from the
multiplicative unit torus. For such actions it is known that the limit for the
growth rate of periodic points exists and equals the entropy of the action. We
extend this result to actions for which the complex variety intersects the
multiplicative torus in a finite set. The main technical tool is the use of
homoclinic points which decay rapidly enough to be summable.Comment: 17 page
On approximate pattern matching for a class of Gibbs random fields
We prove an exponential approximation for the law of approximate occurrence
of typical patterns for a class of Gibssian sources on the lattice
, . From this result, we deduce a law of large numbers and
a large deviation result for the waiting time of distorted patterns.Comment: Published at http://dx.doi.org/10.1214/105051605000000827 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
On the Variational Principle for Generalized Gibbs Measures
We present a novel approach to establishing the variational principle for
Gibbs and generalized (weak and almost) Gibbs states. Limitations of a
thermodynamical formalism for generalized Gibbs states will be discussed. A new
class of intuitively weak Gibbs measures is introduced, and a typical example
is studied. Finally, we present a new example of a non-Gibbsian measure arising
from an industrial application.Comment: To appear in Markov Processes and Related Fields, Proceedings
workshop Gibbs-nonGibb
VARIATIONAL PRINCIPLE FOR FUZZY GIBBS MEASURES
In this paper we study a large class of renormalization transformations of measures on lattices. An image of a Gibbs measure under such transformation is called a fuzzy Gibbs measure. Transformations of this type and fuzzy Gibbs measures appear naturally in many fields. Examples include the hidden Markov processes (HMP), memory-less channels in information theory, continuous block factors of symbolic dynamical systems, and many renormalization transformations of statistical mechanics. The main result is the generalization of the classical variational principle of Dobrushin-Lanford-Ruelle for Gibbs measures to the class of fuzzy Gibbs measures
A concentration inequality for interval maps with an indifferent fixed point
For a map of the unit interval with an indifferent fixed point, we prove an
upper bound for the variance of all observables of variables
which are componentwise Lipschitz. The proof is based on
coupling and decay of correlation properties of the map. We then give various
applications of this inequality to the almost-sure central limit theorem, the
kernel density estimation, the empirical measure and the periodogram.Comment: 26 pages, submitte
Exponential distribution for the occurrence of rare patterns in Gibbsian random fields
We study the distribution of the occurrence of rare patterns in sufficiently
mixing Gibbs random fields on the lattice , . A typical
example is the high temperature Ising model. This distribution is shown to
converge to an exponential law as the size of the pattern diverges. Our
analysis not only provides this convergence but also establishes a precise
estimate of the distance between the exponential law and the distribution of
the occurrence of finite patterns. A similar result holds for the repetition of
a rare pattern. We apply these results to the fluctuation properties of
occurrence and repetition of patterns: We prove a central limit theorem and a
large deviation principle.Comment: To appear in Commun. Math. Phy
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