05-177 Oliver Matte

Supersymmetric Dirichlet operators, spectral gaps, and correlations (433K, Postscript) May 13, 05

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Abstract. In this article we construct a supersymmetric extension of the Dirichlet operator associated with a tempered Gibbs measure on $\RR^{\ZZ^d}$. 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 single-site potentials have a non-degenerated minimum and no other critical point we prove that, for sufficiently large inverse tempertures, 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 Sj\"{o}strand for finite-dimensional Gibbs measures. In the appendix we add a remark on the uniqueness of tempered Gibbs measures at low temperatures.

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