This book provides a mathematically rigorous introduction of the Witten Laplacian methods in Statistical Mechanics. The method provides a new point of view, based on PDE techniques, to approach the problem of computing and estimating thermodynamic functions in classical continuous spin models. The method can be thought as a stronger and more flexible version of the Brascamp-Lieb inequalities and is based on an exact representation of the thermodynamic functions in terms of solutions to a second order partial differential equation, involving a deformation of the standard Laplace-Beltrami operator. The formula was initially introduced by Bernard Helffer and Johanne Sjostrand. The book also provides a complete discussion of the L^2-Theory for the Witten Laplacian equations on zero and one forms. A detailed proof of the exponential decay of the n-point correlation functions is given, along with a new formula suitable for a direct proof of the analyticity of the pressure for certain unbounded models in Statistical Mechanics and Euclidean Field theory.

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