Helffer B.
Splitting in large dimension and infrared estimates II - Moment
inequalities.
(64K, LATEX)
ABSTRACT. This is the continuation of notes written for the NATO-ASI conference
in Il Ciocco (Sept. 96) consisting in the analysis of the links between estimating the
splitting between the two first eigenvalues for the
Schr\"odinger operator $H$ and the proof of infrared estimates for quantities attached
to Gaussian type measures. These notes were mainly reporting on the ``old''
contributions
of Dyson, Fr\"ohlich, Glimm, Jaffe, Lieb, Simon, Spencer (in the
seventies) in
connection with more recent
contributions of Pastur, Khoruzhenko, Barbulyak, Kondratev which
treat in general more sophisticated models. Here we concentrate on the simplest
model related to field theory and extend the
results of Barbulyak-Kondratev by mixing ideas coming from
Pastur-Khozurenko
related to the use of Bogolyubov's inequality with classical
inequalities due to Ginibre, Lebowitz, Sokal.... or in the case when
the temperature $T$ is zero by applying rather elementary
estimates
on Schr\"odinger operators, in order to find lower bounds for
second order moments attached to the measure $\phi \mapsto \Tr \phi
\exp - \beta H/\tr \exp - \beta H$ with $\beta=\frac 1T$. This
question
was
``left to the reader'' in lectures given by J. Fr\"ohlich in 1976
\cite{Fr}, but we think that it is worthwhile to do this ``home work''
carefully.