Decorrelation of a class of Gibbs particle processes and asymptotic properties of <i>U</i>-statistics

Beneš, Viktor; Hofer-Temmel, Christoph; Last, Günter; Vecera, Jakub

doi:10.1017/jpr.2020.51

Cited by 14 publications

(14 citation statements)

References 30 publications

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“…Note that the interaction between the points of such a process depends purely on the corresponding clusters and that this interaction need not have a finite range. The uniqueness result we provide in this setting covers the results by Hofer-Temmel and and Beneš et al (2020), who consider particle processes with the binary relation on the state space given through the intersection of particles, but is much more general. Also we ensure existence of the processes from Hofer-Temmel and within the region of uniqueness.…”

Section: Introductionsupporting

confidence: 70%

“…Moreover, our approach leads to manageable conditions for the existence (and uniqueness) of Gibbsian particle processes, detailed in Section 10. Though the existence result is substantially more general, it particularly covers the conjecture by Beneš et al (2020) who study a special class of such particle processes and emphasize that their existence is not guaranteed by the available literature.…”

Section: Introductionmentioning

confidence: 78%

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Structural properties of Gibbsian point processes in abstract spaces

Betsch¹

2022

Preprint

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In the language of random counting measures many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with cluster-dependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.

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Section: Introductionsupporting

confidence: 70%

Section: Introductionmentioning

confidence: 78%

Structural properties of Gibbsian point processes in abstract spaces

Betsch¹

2022

Preprint

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“…Disagreement percolation provides an expansion-free route to proofs of uniqueness and exponential mixing for Gibbs measures; see [25, Chapter 7] for Gibbs measures on lattices and [29] and [30] as well as [14, Section 2.7] for models in , and [2] for Gibbs particle processes. It is an interesting question which approach – percolation or convergence of cluster expansions – trumps the other as far as proving uniqueness of the infinite-volume Gibbs measure goes.…”

Section: Resultsmentioning

confidence: 99%

Cluster Expansions for GIBBS Point Processes

Jansen

2019

Adv. Appl. Probab.

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We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández and Procacci (2010). In addition, we provide formulas for double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

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“…The corresponding Gibbs model are hard particles in equilibrium, while the case of a general V could be addressed as soft particles in equilibrium, at least if V is translation invariant. Theorem 1.1 requires (ϕ, λ) to be subcritical, while the previous results from [16,1] (when specialized to non-negative pair potentials) require the Boolean model (ϕ ∞ , λ) to be subcritical. Since ϕ(K, L) ≤ ϕ ∞ (K, L) our result gives better bounds on the uniqueness region.…”

Section: Comments and Examplesmentioning

confidence: 96%

“…Uniqueness follows if the so-called ancestor clans, coming from an embedding into a space-time Poisson process, are finite. The resulting bounds on the domain of uniqueness are not explicit but might be comparable with the branching bounds (see also the discussion in [1]).…”

Section: Introductionmentioning

confidence: 97%

On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential

Betsch¹,

Last²

2021

Preprint

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We prove that the distribution of a Gibbs process with non-negative pair potential is uniquely determined as soon as an associated Poisson-driven random connection model (RCM) does not percolate. Our proof combines disagreement coupling in continuum (established in [16, 23]) with a coupling of a Gibbs process and a RCM. The improvement over previous uniqueness results is illustrated both in theory and simulations.

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Decorrelation of a class of Gibbs particle processes and asymptotic properties of U-statistics

Cited by 14 publications

References 30 publications

Structural properties of Gibbsian point processes in abstract spaces

Structural properties of Gibbsian point processes in abstract spaces

Cluster Expansions for GIBBS Point Processes

On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential

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