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Beschreibung:
This thesis deals with the statistical mechanics of lattice models. It has two main contributions. On the one hand we implement a general framework for a rigorous renormalisation group approach to gradient models. This approach relies on work by Bauerschmidt, Brydges, and Slade and extends earlier results for gradient interface models by Adams, Kotecký and Müller. On the other hand we use those results to analyse microscopic models for discrete elasticity at small positive temperature and in particular prove convexity properties of the free energy. The first Chapter is introductory and discusses the necessary mathematical background and the physical motivation for this thesis. Chapters 2 to 4 then contain a complete and almost self contained implementation of the renormalisation group approach for gradient models. Chapter 2 is concerned with a new construction of a finite range decomposition with improved regularity. Finite range decompositions are an important ingredient in the renormalisation group approach but also appear at various other places. The new finite range decomposition helps to avoid a loss of regularity and several technical problems that were present in the earlier applications of the renormalisation group technique to gradient models. In the third Chapter we analyse generalized gradient models and discrete models for elasticity and we state our main results: At low temperatures the surface tension is locally uniformly convex and the scaling limit is Gaussian. Moreover, we show that those statements can be reduced to a general statement about perturbations of massless Gaussian measures using suitable null Lagrangians. This is a first step towards a mathematical understanding of elastic behaviour of crystalline solids at positive temperatures starting from microscopic models. The fourth Chapter contains the renormalisation group analysis of gradient models. The main result is a bound for certain perturbations of Gaussian gradient measures that implies the results of the previous chapters. This ...