Laplace operators in deRham complexes associated with measures on configuration spaces

Albeverio, Sergio; Daletskii, Alexei; Kondratiev, Yuri; Lytvynov, Eugene

doi:10.1016/s0393-0440(02)00221-8

Cited by 12 publications

(17 citation statements)

References 42 publications

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“…The results of the latter reference have been generalized in [11,12]. A differential form calculus on configuration spaces is used, e.g., in [3]. The Helffer-Sjöstrand formula for the correlation has also been used in an infinite-dimensional setting before.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric Dirichlet Operators, Spectral Gaps, and Correlations

Matte

2006

Ann. Henri Poincaré

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In this article we construct a supersymmetric extension of the Dirichlet operator associated with a tempered Gibbs measure on R Z d . Under fairly general assumptions on the interaction potentials we show that the Dirichlet operator (resp. its supersymmetric extension) is essentially selfadjoint on the set of smooth, bounded cylinder functions (resp. differential forms), for all inverse temperatures. Assuming that the on-site potentials have a non-degenerated minimum and no other critical point we prove that, for sufficiently large inverse temperatures, one observes a number of subsequent gaps in the spectrum of the Dirichlet operator. For translation invariant systems with a sufficiently weak (but in general infinite range) ferromagnetic interaction, we prove the validity of a formula for the leading asymptotics of the correlation of two spin variables, as their distance and the inverse temperature tend to infinity, which has originally been derived by J. Sjöstrand for finite-dimensional systems.

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Section: Introductionmentioning

confidence: 99%

Supersymmetric Dirichlet Operators, Spectral Gaps, and Correlations

Matte

2006

Ann. Henri Poincaré

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“…From the physical point of view, this describes a passage from a system of particles without interaction (free gas) to an interacting particle system, see [11] and references within. For a wide class of measures, including Gibbs measures of Ruelle type and Gibbs measures in low activity-high temperature regime, the de Rham complex has been introduced and studied in [5]. The structure of the corresponding Laplacian is much more complicated in this case, and the spaces of harmonic forms have not been studied yet.…”

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confidence: 99%

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Albeverio¹,

Daletskii²

2006

Publ. Res. Inst. Math. Sci.

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The space Γ X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ X , equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.

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“…Measures of such type appear, via the generalized Mecke identity, in the theory of configuration spaces, and in particular in the theory of Laplace operators on differential forms over Γ X (see [5][6][7]). In fact, the Witten Laplacian H associated with σ is a "part" of the Hodgede Rham operator on Γ X associated with the Gibbs measure μ.…”

Section: Introductionmentioning

confidence: 99%