On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

Zur lexikographischen Darstellung der Adjektivsyntax in neueren einsprachigen Wörtbüchern des Deutschen

The main objects of the investigation were the syntactic functions of adjectives. The reason for the interest in these functions are the different modes of use, in which an adjective can occur. All together an adjective can take three different modes of use: attributive (e. g. a fast car), predicative (e. g. the car is fast) and adverbial (e. g. the car drives fast). Since an adjective cannot always take every function, some dictionaries (esp. learner s dictionaries) deliver information within the lexical entry about any restrictions. The purpose of the research consisted of a comparison in relation to the lexical entries of adjectives, which were investigated within four selected monolingual German-speaking dictionaries. The comparison of the syntactical data of adjectives were done to work out the differences and the common characteristics of the lexical entries concerning the different modes of use and to analyse respective to assess them. In the foreground, however, were the differences of the syntactical information. Concerning those differences it had to be worked out, which entry is the grammatically right one respective if one entry is in fact wrong. To find that out an empirical analysis was needed, which based on the question in which way an adjective is used within a context as far as there are no conforming data within the dictionaries. The delivery of the correctness and the homogeneity of lexical entries of German-speaking dictionaries are very important to support people who are learning the German language and to ensure the user friendliness of dictionaries. Throughout the investigations it became clear that in almost half of the cases (over 40 %) syntactical information of adjectives differ from each other within the dictionaries. These differences make it for non-native speakers of course very difficult to understand the correct usage of an adjective. Thus the main aim of the doctoral thesis was it to deliver and to demonstrate the clear syntactical usage of a certain amount of adjectives.