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Ferromagnetic integrals, correlations and maximum principles. (English) Zbl 0831.35031
In a previous paper [Astérisque, 210, 303-326 (1992; Zbl 0796.35123)] the author developed an idea of I. M. Singer, B. Wong, S.-T. Yau and S. S.-T. Yau [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 319-333 (1985; Zbl 0603.35070)] to use the maximum principle in the study of the logarithm of the first eigenfunction of a Schrödinger operator on \(\mathbb{R}^m\) with a strictly convex potential, and for suitable sequences of potentials, he was able to establish the exponential convergence of the first eigenvalue divided by \(m\), when \(m\) tends to infinity. More recently, the reviewer and the author [J. Statistical Phys. 74, No. 1-2, 349-369 (1994)] employed a similar method to study expectation values of the form: \[ \langle f \rangle = \left. \left( \int_{\mathbb{R}^m} \exp - {\varphi (x) \over h} f(x) dx \right) \right/ \int_{\mathbb{R}^m} \exp - {\varphi (x) \over h} dx, \] and in particular the correlations \(({\mathcal C})\) \(\langle (x_j - \langle x_j \rangle) (x_k - \langle x_k \rangle) \rangle\), for large \(m\) and \(|j - k |\). Under suitable assumptions, implying uniform strict convexity for \(\varphi\), it was proved that \(({\mathcal C})\) can be estimated by \({\mathcal O} (1) \exp - |j - k |/C\). In this paper, the author improves the use of the maximum principle and as an application considers for the correlations \(({\mathcal C})\) a critical case and proves power decay upper bounds in terms of the fundamental solution of a certain elliptic operator. In the last part, the author formulates a general maximum principle and gives two applications.
Reviewer: B.Helffer (Orsay)

MSC:
35C20 Asymptotic expansions of solutions to PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P15 Estimates of eigenvalues in context of PDEs
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References:
[1] H.J. BRASCAMP, E.H. LIEB, On extensions of the brunn-minkovski and Prékopa Leindler theorems,..., J. Funct. An., 22 (1976), 366-389. · Zbl 0334.26009
[2] J. BRICMONT, J.R. FONTAINE, J.L. LEBOWITZ, T. SPENCER, Lattice systems with continuous symmetry II. decay of correlations, Comm. Math. Phys., 78 (1981), 363-373.
[3] J. BRICMONT, J.R. FONTAINE, J.L. LEBOWITZ, E.H. LIEB, T. SPENCER, Lattice systems with continuous symmetry III. low temperature asymptotic expansion for the plane rotator model, Comm. Math. Phys., 78 (1981), 545-566.
[4] P. CARTIER, Inégalités de correlation en mécanique statistique, Sém. Bourbaki, 25ème année, 1972-1973, n° 431, Springer LNM n° 383.
[5] R.S. ELLIS, Entropy, large deviations and statistical mechanics, Grundlehren der Math. Wiss., 271, Springer (1985). · Zbl 0566.60097
[6] J. GLIMM, A. JAFFEE, Quantum physics, a functional integral point of view, second edition, Springer (1987).
[7] G. GRIMMETT, Percolation, Springer (1989). · Zbl 0691.60089
[8] F. GUERRA, L. ROSEN, B. SIMON, The p(ø)2 Euclidean quantum field theory as classical statistical mechanics, Ann. Math., 101 (1975), 111-259.
[9] B. HELFFER, J. SJÖSTRAND, On the correlation for Kac like models in the convex case, report n° 9, 1992-1993, Institut Mittag-Leffler. · Zbl 0946.35508
[10] I.M. SINGER, B. WONG, S.T. YAU, S.S.T. YAU, An estimate of the gap of the first two eigenvalues of the Schrödinger operator, Ann. Sc. Norm. Sup. Pisa (ser. 4), 12 (1985), 319-333. · Zbl 0603.35070
[11] J. SJÖSTRAND, Exponential convergence of the first eigenvalue divided by the dimension, for certain sequences of Schrödinger operators, Astérisque, 210 (1992), 303-326. · Zbl 0796.35123
[12] A.D. SOKAL, Mean field bounds and correlation inequalities, J. Stat. Phys., 28 (1982), 431-439.
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