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Published in last 50 years
We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.
We study classical lattice systems, in particular real spin glasses with Ruderman-Kittel interactions and dipole gases, with interactions of very long (non-summable) range but variable sign. Using the Kac-Siegert representation of such systems and Brascamp-Lieb inequalities we are able to establish detailed properties of the high-temperature phase, such as decay of connected correlations, for these systems.
We consider models, with an abelian continuous group of symmetry, of the type: $$H = \sum\limits_x {\left[ {\frac{1}{2}(\nabla _x \phi )^2 + \frac{\lambda }{4}(\nabla _x \phi )^4 } \right].}$$ We generalize Brascamp-Lieb inequalities to get (λ-independent) bounds on the low momentum behaviour of general correlation functions when these are truncated into two clusters. We then use this result to derive an asymptotic expansion (up the second order in λ) of the dielectric constant of this system.