Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

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.

Spirometric studies on the adult general population of Helsinki : bronchodilation responses, determinants, and intrasession repeatability of FEV1, FEV6, FVC, and forced expiratory time

Spirometry is the most widely used lung function test in the world. It is fundamental in diagnostic and functional evaluation of various pulmonary diseases. In the studies described in this thesis, the spirometric assessment of reversibility of bronchial obstruction, its determinants, and variation features are described in a general population sample from Helsinki, Finland. This study is a part of the FinEsS study, which is a collaborative study of clinical epidemiology of respiratory health between Finland (Fin), Estonia (Es), and Sweden (S). Asthma and chronic obstructive pulmonary disease (COPD) constitute the two major obstructive airways diseases. The prevalence of asthma has increased, with around 6% of the population in Helsinki reporting physician-diagnosed asthma. The main cause of COPD is smoking with changes in smoking habits in the population affecting its prevalence with a delay. Whereas airway obstruction in asthma is by definition reversible, COPD is characterized by fixed obstruction. Cough and sputum production, the first symptoms of COPD, are often misinterpreted for smokers cough and not recognized as first signs of a chronic illness. Therefore COPD is widely underdiagnosed. More extensive use of spirometry in primary care is advocated to focus smoking cessation interventions on populations at risk. The use of forced expiratory volume in six seconds (FEV6) instead of forced vital capacity (FVC) has been suggested to enable office spirometry to be used in earlier detection of airflow limitation. Despite being a widely accepted standard method of assessment of lung function, the methodology and interpretation of spirometry are constantly developing. In 2005, the ATS/ERS Task Force issued a joint statement which endorsed the 12% and 200 ml thresholds for significant change in forced expiratory volume in one second (FEV1) or FVC during bronchodilation testing, but included the notion that in cases where only FVC improves it should be verified that this is not caused by a longer exhalation time in post-bronchodilator spirometry. This elicited new interest in the assessment of forced expiratory time (FET), a spirometric variable not usually reported or used in assessment. In this population sample, we examined FET and found it to be on average 10.7 (SD 4.3) s and to increase with ageing and airflow limitation in spirometry. The intrasession repeatability of FET was the poorest of the spirometric variables assessed. Based on the intrasession repeatability, a limit for significant change of 3 s was suggested for FET during bronchodilation testing. FEV6 was found to perform equally well as FVC in the population and in a subgroup of subjects with airways obstruction. In the bronchodilation test, decreases were frequently observed in FEV1 and particularly in FVC. The limit of significant increase based on the 95th percentile of the population sample was 9% for FEV1 and 6% for FEV6 and FVC; these are slightly lower than the current limits for single bronchodilation tests (ATS/ERS guidelines). FEV6 was proven as a valid alternative to FVC also in the bronchodilation test and would remove the need to control duration of exhalation during the spirometric bronchodilation test.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.