Renormalisation in discrete elasticity

Abstract

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.<br /> The first Chapter is introductory and discusses the necessary mathematical background and the physical motivation for this thesis.<br /> Chapters 2 to 4 then contain a complete and almost self contained implementation of the renormalisation group approach for gradient models. <br /> 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. <br /> 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.<br /> 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 generalizes earlier results for scalar nearest neighbour models to vector-valued finite range interactions. We also require a much weaker growth assumption for the perturbation. This is possible because we introduce a new solution to the large field problem based on an alternative construction of the weight functions using Gaussian calculus.<br /> The last Chapter has a slightly different focus. We investigate gradient interface models for a specific class of non-convex potentials for which phase transitions occur in dimension two. The analysis of these potentials is based on the relation to a random conductance model. We study properties of this random conductance model and in particular prove correlation inequalities and reprove the phase transition result relying on planar duality instead of reflection positivity.