Oliver Matte
Supersymmetric Dirichlet operators, spectral gaps, and correlations
(433K, Postscript)
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.