Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics
World Scientific, 2002 - 179 Seiten
This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log -- Sobolev inequality.
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Witten Laplacians approach
Problems in statistical mechanics with
Laplace integrals and transfer operators
Semiclassical analysis for the transfer
Basic facts in spectral theory and on
1-forms A C Zd adjoint analyze associated assumption Bakry-Emery Cauchy-Schwarz Chapter compact resolvent computation condition consequently context convergence corresponding covariance decay estimates defined denote DeStl differential domain eigenfunction eigenvalue eigenvector equation essential spectrum essentially selfadjoint example Exercise exp(u(t explicit formula Friedrichs extension function given gives harmonic approximation harmonic oscillator Hess Hilbert space identity introduce Ising model Laplace integrals largest eigenvalue Lemma lower bound lowest eigenvalue measure minima Note observe obtain orthonormal basis parameters particular Perron-Frobenius Theorem phase Poincare inequality presented problem proof prove quadratic recall Remark riemannian manifold satisfies Schrodinger operator Section selfadjoint extension selfadjoint operator semi-classical analysis single spin Sjostrand Sobolev Sobolev space spectral spectral theory statistical mechanics strictly positive sublemma symmetric operator tends to oo Theorem theory thermodynamic limit trace class transfer matrix transfer operator uniform decay upper bound variable Witten Laplacian xmin