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83
Invariance principle for reversible Markov processes with application to diffusion in the percolation regime
, 1985
"... We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the inf ..."
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Cited by 90 (5 self)
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We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the twodimensional bond percolation model, (ii) for a ddimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in a ddimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in a ddimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.
The spectral gap for a Glaubertype dynamics in a continuous gas
, 2000
"... . We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenter ..."
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Cited by 38 (6 self)
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. We consider a continuous gas in a d dimensional rectangular box with a nite range, positive pair potential, and we construct a Markov process in which particles appear and disappear with appropriate rates so that the process is reversible w.r.t. the Gibbs measure. If the thermodynamical paramenters are such that the Gibbs specication satises a certain mixing condition, then the spectral gap of the generator is strictly positive uniformly in the volume and boundary condition. The required mixing condition holds if, for instance, there is a convergent cluster expansion. Key Words: Spectral gap, Gibbs measures, continuous systems, birth and death processes Mathematics Subject Classication: 82C21, 60K35, 82C22, 60J75 This work was partially supported by GNAFA and by \Conanziamento Murst" v1.4 1. Introduction We consider a continuous gas in a bounded volume R d , distributed according the Gibbs probability measure associated to a nite range pair potential '. The Gibbs measu...
Point processes for unsupervised line network extraction in remote sensing
 IEEE Trans. Pattern Anal. Mach. Intell
, 2005
"... Abstract—This paper addresses the problem of unsupervised extraction of line networks (for example, road or hydrographic networks) from remotely sensed images. We model the target line network by an object process, where the objects correspond to interacting line segments. The prior model, called “Q ..."
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Cited by 28 (5 self)
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Abstract—This paper addresses the problem of unsupervised extraction of line networks (for example, road or hydrographic networks) from remotely sensed images. We model the target line network by an object process, where the objects correspond to interacting line segments. The prior model, called “Quality Candy, ” is designed to exploit as fully as possible the topological properties of the network under consideration, while the radiometric properties of the network are modeled using a data term based on statistical tests. Two techniques are used to compute this term: one is more accurate, the other more efficient. A calibration technique is used to choose the model parameters. Optimization is done via simulated annealing using a Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm. We accelerate convergence of the algorithm by using appropriate proposal kernels. The results obtained on satellite and aerial images are quantitatively evaluated with respect to manual extractions. A comparison with the results obtained using a previous model, called the “Candy ” model, shows the interest of adding quality coefficients with respect to interactions in the prior density. The relevance of using an offline computation of the data potential is shown, in particular, when a proposal kernel based on this computation is added in the RJMCMC algorithm. Index Terms—Stochastic processes, Monte Carlo, simulated annealing, edge and feature detection, remote sensing. æ 1
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 28 (11 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].
Phase Transition in Continuum Potts Models
 Commun. Math. Phys
, 1996
"... We establish phase transitions for a class of continuum multitype particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (betw ..."
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Cited by 24 (8 self)
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We establish phase transitions for a class of continuum multitype particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a randomcluster representation analogous to the FortuinKasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with superstable interaction. 1 Introduction Although the study of phase transitions for Gibbsian systems is one of the main subjects of statistical mechanics, examples of models exhibiting phase transition are mainly restricted to lattice systems. For syste...
Diffeomorphism Groups And Current Algebras: configuration space analysis in quantum theory
 PREPRINT 97073, SFB 343
, 1997
"... The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the fr ..."
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Cited by 16 (1 self)
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The constuction of models of nonrelativistic quantum fields via current algebra representations is presented using a natural differential geometry of the configuration space \Gamma of particles, the corresponding classical Dirichlet operator associated with a Poisson measure on \Gamma, being the free Hamiltonian. The case with interactions is also discussed together with its relation to the problem of unitary representations of the diffeomorphism group on R^d.
Alexander: Traces of Semigroups Associated with Interacting Particle Systems
"... gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems ..."
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Cited by 14 (3 self)
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gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2006 Traces of semigroups associated with interacting particle systems
Analysis and geometry on configuration spaces: The Gibbsian case
, 1998
"... Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Ga ..."
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Cited by 11 (3 self)
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Using a natural "Riemanniangeometrylike" structure on the configuration space \Gamma over IR d , we prove that for a large class of potentials OE the corresponding canonical Gibbs measures on \Gamma can be completely characterized by an integration by parts formula. That is, if r \Gamma is the gradient of the Riemannian structure on \Gamma one can define a corresponding divergence div \Gamma OE such that the canonical Gibbs measures are exactly those measures ¯ for which r \Gamma and div \Gamma OE are dual operators on L 2 (\Gamma; ¯). One consequence is that for such ¯ the corresponding Dirichlet forms E \Gamma ¯ are defined. In addition, each of them is shown to be associated with a conservative diffusion process on \Gamma with invariant measure ¯. The corresponding generators are extensions of the operator \Delta \Gamma OE := div \Gamma OE r \Gamma . The diffusions can be characterized in terms of a martingale problem and they can be considered as a Brown...
Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction
 PROBABILITY THEORY AND RELATED FIELDS
, 1994
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