97-139 Helffer B.

Splitting in large dimension and infrared estimates II - Moment inequalities. (64K, LATEX) Mar 20, 97

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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.

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