The Simon-Lieb inequality for the \(\phi ^ 4\)-theory in two dimensions
\textit{D. Brydges} and \textit{T. Spencer} [Commun. Math. Phys. 83, 125-152 (1982)], established a Simon-Lieb inequality for the \(\phi^ 4\)-measure in d-dimensional lattice. The paper under review shows its two- dimensional continuum version: If D is a bounded domain in \({\mathbb{R}}^ 2\) with smooth boundary \(\partial D\), then for a.e. \(x\in D\) and \(y\in \bar D^ c\), \[ 0\leq <\phi_ x\phi_ y>\leq \int_{\partial D}S(du)\frac{\partial <{\tilde \phi}_ x{\tilde \phi}_ u>_{\tilde H}}{\partial n_ u}<\phi_ u\phi_ y>, \] where S(du) is the line element of \(\partial D\), and \(\partial /\partial n_ u\) is the inward normal derivative at \(u\in \partial D\). Further \(<>\) denotes the expectation of \(\phi^ 4\)-measure, and \(<>_{\tilde H}\) the expectation of \(\phi^ 4\)-measure with the half-Dirichlet boundary conditions on D, both with spatial cutoff. The proof uses \textit{E. B. Dynkin}'s theorem [J. Funct. Anal. 55, 344-376 (1984; Zbl 0533.60061)] on Symanzik's representations and its extension given also in the paper.
P(\(\phi \) )-(constructive) field theory
scientific article; zbMATH DE number 4080190
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