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The semigroup of the Glauber dynamics of a continuous system of free particles
, 2004
"... We study properties of the semigroup (e −tH)t≥0 on the space L 2 (ΓX,π), where ΓX is the configuration space over a locally compact second countable Hausdorff topological space X, π is a Poisson measure on ΓX, and H is the generator of the Glauber dynamics. We explicitly construct the corresponding ..."
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We study properties of the semigroup (e −tH)t≥0 on the space L 2 (ΓX,π), where ΓX is the configuration space over a locally compact second countable Hausdorff topological space X, π is a Poisson measure on ΓX, and H is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels (Pt)t≥0 and, using it, we prove the main results of the paper: the Feller property of the semigroup (Pt)t≥0 with respect to the vague topology on the configuration space ΓX, and the ergodic property of (Pt)t≥0. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every γ ∈ ΓX, will never leave ΓX and has cadlag sample paths in ΓX.
Boundary Conditions and Mixing Time
 In Proceedings of the Forty Fourth Annual Symposium on Foundations of Computer Science
, 2003
"... We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an nvertex regular tree with (+)boundary remains O(n log n) at all temperatures (in contrast to the free bound ..."
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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics for the Ising model. Specifically, we show that the mixing time on an nvertex regular tree with (+)boundary remains O(n log n) at all temperatures (in contrast to the free boundary case, where the mixing time is not bounded by any fixed polynomial at low temperatures) . We also show that this bound continues to hold in the presence of an arbitrary external field. Our results are actually stronger, and provide tight bounds on the logSobolev constant and the spectral gap of the dynamics. In addition, our methods yield simpler proofs and stronger results for the mixing time in the regime where it is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard constraints like the antiferromagnetic Potts model at zero temperature (colorings) and the hardcore model (independent sets).
On convergence of dynamics of hopping particles to a birthanddeath process in continuum
, 2008
"... We show that some classes of birthanddeath processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics) ..."
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We show that some classes of birthanddeath processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)
Author manuscript, published in "Journal of Mathematical Imaging and Vision (2009)" Object
, 2008
"... extraction using a stochastic birthanddeath dynamics in continuum ..."
unknown title
, 2005
"... Spectral gap estimates for interacting particle systems via a Bochner–type identity ..."
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Spectral gap estimates for interacting particle systems via a Bochner–type identity
unknown title
, 2005
"... Spectral gap estimates for interacting particle systems via a Bochner–type identity ..."
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Spectral gap estimates for interacting particle systems via a Bochner–type identity
unknown title
, 709
"... On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum E. Lytvynov and P.T. Polara ..."
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On convergence of generators of equilibrium dynamics of hopping particles to generator of a birthanddeath process in continuum E. Lytvynov and P.T. Polara
SLOW DECAY OF GIBBS MEASURES WITH HEAVY TAILS
, 811
"... Abstract. We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails. Spins are unbounded. The interactions are bounded and finite range. The self potential enters into two classes of measures, κconcave probability measure and subexponential laws, for which it is known ..."
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Abstract. We consider Glauber dynamics reversible with respect to Gibbs measures with heavy tails. Spins are unbounded. The interactions are bounded and finite range. The self potential enters into two classes of measures, κconcave probability measure and subexponential laws, for which it is known that no exponential decay can occur. We prove, using coercive inequalities, that the associated infinite volume semigroup decay to equilibrium polynomially and stretched exponentially, respectively. Thus improving and extending previous results by Bobkov and Zegarlinski. 1.
Invariant measures for Glauber dynamics of continuous systems
, 2003
"... We consider Glaubertype stochastic dynamics of continuous systems [BCC02], [KL03], a particular case of spatial birthanddeath processes. The dynamics is defined by a Markov generator in such a way that Gibbs measures of Ruelle type are symmetrizing, and hence invariant for the stochastic dynamics ..."
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We consider Glaubertype stochastic dynamics of continuous systems [BCC02], [KL03], a particular case of spatial birthanddeath processes. The dynamics is defined by a Markov generator in such a way that Gibbs measures of Ruelle type are symmetrizing, and hence invariant for the stochastic dynamics. In this work we show that the converse statement is also true. Namely, all invariant measures satisfying Ruelle bound condition are grand canonical Gibbsian for the potential defining the dynamics. The proof is based on the observation that the wellknown KirkwoodSalsburg equation for correlation functions is indeed an equilibrium equation for the stochastic dynamics. 1 1