Complete Asymptotics for Correlations of Laplace Integrals in the Semi-classical LimiCot

The book summarizes progress achieved by the author and others (V. Bach, T. Jecko) in the study of the correlation decay for Laplace integrals of the type ∫ exp{- φ(x)/h}dx in the low temperature regime, i.e., for small values of h. The variable x is taken from RΛ, where Λ is either a d-dimensional torus, or a finite subset of an infinite lattice. The function φ(x) is assumed to satisfy a certain condition of positive definiteness for its second derivative near its (unique) minimum point. The case of “one potential well”, with “nearly quadratic” behaviour around the minimum point is thus considered. The main result of the book is the following: A detailed formula for the pair correlation function is given as an expansion in the variable h, (the expectation < > is given in the above mentioned Gibbs measure). The result is rather general and greatly improves earlier results, due to the author and others. The essential tool used in the proof, introduced already in the previous works by Helffer and Sjöstrand, is the formula expressing the pair correlation function in terms of the “Witten Laplace” operator (defined as the Hodge Laplacian with the help of the corresponding creation/ annihilation operators). The problem is then translated to the study of spectral properties (bottom of the spectrum) of such an operator, more precisely of its zeroth and first part, applying a method called the “Grushin (Feshbach) approach” by the author. The book is divided into twelve short chapters and two appendices. Chapter 0 and 1 introduce the problem and formulate the main result (and its generalization); the remainder develops the details of various steps of the proof (reshuffling of the creation/ annihilation operators, investigation of higher order Grushin problems, asymptotics of the correlations, etc.). The method developed in the presented book is an important contribution to a subject which has great importance in statistical physis and quantum field theory and which was up to now treated in the literature by quite different methods, like cluster expansions

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ISBN 2-85629-097-3
FRF 150