zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Witten Laplacian method for the decay of correlations. (English) Zbl 1144.82003
Summary: The aim of this paper is to apply direct methods to the study of integrals that appear naturally in Statistical Mechanics and Euclidean Field Theory. We provide weighted estimates leading to the exponential decay of the two-point correlation functions for certain classical convex unbounded models. The methods involve the study of the solutions of the Witten Laplacian equations associated with the Hamiltonian of the system.

MSC:
82B05Classical equilibrium statistical mechanics (general)
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
References:
[1]Antoniouk, A.V., Antoniouk, A.V.: Decay of correlations and uniqueness of Gibbs lattice systems with nonquadratic interaction. J. Math. Phys. 37(11) (1996)
[2]Bach, V., Moller, J.S.: Correlation at low temperature, exponential decay. J. Funct. Anal. 203, 93–148 (2003) · Zbl 1031.82003 · doi:10.1016/S0022-1236(03)00046-6
[3]Bach, V., Jecko, T., Sjöstrand, J.: Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature. Ann. Henri Poincare 1, 59–100 (2000) · Zbl 1021.82002 · doi:10.1007/PL00001002
[4]Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Kluwer Academic, Dordrecht (1991)
[5]Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems including inequalities for log concave functions, and with application to the diffusion equation. J. Funct. Anal. 22, 366–389 (1976) · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5
[6]Evans, L.C.: Partial Differential Equations. Am. Math. Soc., Providence (1998)
[7]Helffer, B., Sjöstrand, J.: On the correlation for Kac-like models in the convex case. J. Stat. Phys. 74(1/2) (1994)
[8]Johnsen, J.: On the spectral properties of Witten-Laplacians, their range of projections and Brascamp-Leib’s inequality. Integr. Equ. Oper. Theory 36, 288–324 (2000) · Zbl 1023.58012 · doi:10.1007/BF01213926
[9]Kac, M.: Mathematical Mechanism of Phase Transitions. Gordon & Breach, New York (1966)
[10]Kneib, J.M., Mignot, F.: Équation de Schmoluchowski généralisée (Generalized Smoluchowski equation). Ann. Mat. Pura Appl. 167(4), 257–298 (1994). · Zbl 0813.35049 · doi:10.1007/BF01760336
[11]Sjöstrand, J.: Exponential convergence of the first eigenvalue divided by the dimension, for certain sequences of Schrödinger operators. Méthodes semi-classiques, vol. 2 (Nantes, 1991). Astérisque 210(10), 303–326 (1992)
[12]Sjöstrand, J.: Correlation asymptotics and Witten Laplacians. Algebra Anal. 8(1), 160–191 (1996)
[13]Troianiello, G.M.: Elliptic Differential Equations and Obstacle Problems. Plenum, New York (1987)
[14]Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)
[15]Yosida, K.: Functional Analysis. Springer Classics in Mathematics. Springer, Berlin (1995)