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Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma

In: Inequalities

Author

Listed:
  • H. J. Brascamp

    (Princeton University, Department of Physics)

  • E. H. Lieb

    (Princeton University, Department of Mathematics and Physics)

Abstract

THE following is a preliminary report on some recent work, the full details of which will be published elsewhere. We have come across some inequalities about integrals and moments of log concave functions which hold in the multidimensional case and which are useful in obtaining estimates for multidimensional modified Gaussian measures. By making a small jump (we shall not go into the technical details) from the finite to the infinite dimensional case, upper and lower bounds to certain types of functional integrals can be obtained. As a non-trivial application of the latter we shall, for the first time, prove that the one-dimensional one-component quantummechanical plasma has long-range order when the interaction is strong enough. In other words, the Wigner lattice can exist, in one dimension at least. As another application we shall prove a log concavity theorem about the fundamental solution (Green’s function) of the diffusion equation.

Suggested Citation

  • H. J. Brascamp & E. H. Lieb, 2002. "Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma," Springer Books, in: Michael Loss & Mary Beth Ruskai (ed.), Inequalities, pages 403-416, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55925-9_34
    DOI: 10.1007/978-3-642-55925-9_34
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