Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma
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
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Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma H. J.
BRASCAMP AND
E. H.
LIEB
1.1. Introduction 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. 1.2. Basic concavity theorem We begin with a theorem (Theorem 1.1) which, to the best of our knowledge, is new and which constitutes the basis of all our other inequalities. DEFINITION 1.1. A function F from R n to R is a log concave function if F(x)~O, VOx ERn, and F(xrF(y)H ~F[Ax +(l-A)Y], VOx, y E R n and A E (0, 1). If the inequality is reversed, we say that F is log convex. We shall sometimes write F(x) = e t (» and f is concave, but it then is understood that f can take on the value - 00. We say that F is even if F(x) = F( - x), VOx. Two important examples of log concave functions are: (a) F(x) = exp[ - (x, Ax)], where A is any symmetric real positivesemidefinite quadratic form on Rn. (b) Let C be any convex set in R n and let XcCx) = 1 for x E C, XcCx) = 0 for x ~ C be the characteristic function of C. Then Xc is a log concave function. Xc is even if and only if C is balanced, i.e. x E C =? - X E C. THEOREM 1.1. Let F be a log concave function on R m + n and F: (x, y) ~ F(x, y) for x E R m, y ERn. Then G(x) == SR" F(x, y) dy is a log concave function on Rm.
We have four different proofs of this theorem, one of which is the following. M. Loss et al. (eds.), Inequalities © Springer-Verlag Berlin Heidelberg 2002
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With H.J. Brascamp in Functional Integration and Its Applications, A.M. Arthurs, ed.
Some inequalities for Gaussian measures and the
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Proof. It is sufficient to prove the theorem when m = n = 1; the general case follows by Fubini's theorem and induction. Choose two points x and x' such that G(x);iO and G(x');iO. We may assume that sup{F(x, y)}=sup{F(x', y)},
for otherwise we can replace F(x, y) by e bx F(x, y) with b suitably chosen. For each z ;;.: 0, define C(z) = {(x, y)IF(x, y);;':z}cR 2 , C(x, y)={yIF(x, y);;,:z}cR and g(x, z)=meas{C(x, z)}. Then (i) C(z) is convex and thus C(x, z) is an interval; (ii) G(x) = J~ g(x, z) dz; (iii) for all O~ A ~ 1, g(Ax + (1- A)x', z);;.: Ag(x, z) +(1- A)g(x', z). This la
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