dc.contributor.author Haug, Nils Adrian dc.date.accessioned 2017-12-18T14:08:30Z dc.date.available 2017-12-18T14:08:30Z dc.date.issued 2017-10-18 dc.date.submitted 2017-12-18T11:15:25.465Z dc.identifier.citation Haug, N.A. 2017. Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers. Queen Mary University of London en_US dc.identifier.uri http://qmro.qmul.ac.uk/xmlui/handle/123456789/30706 dc.description PhD en_US dc.description.abstract The subject of this thesis is the asymptotic behaviour of generating functions en_US of different combinatorial models of two-dimensional lattice walks and polygons, enumerated with respect to different parameters, such as perimeter, number of steps and area. These models occur in various applications in physics, computer science and biology. In particular, they can be seen as simple models of biological vesicles or polymers. Of particular interest is the singular behaviour of the generating functions around special, so-called multicritical points in their parameter space, which correspond physically to phase transitions. The singular behaviour around the multicritical point is described by a scaling function, alongside a small set of critical exponents. Apart from some non-rigorous heuristics, our asymptotic analysis mainly consists in applying the method of steepest descents to a suitable integral expression for the exact solution for the generating function of a given model. The similar mathematical structure of the exact solutions of the different models allows for a unified treatment. In the saddle point analysis, the multicritical points correspond to points in the parameter space at which several saddle points of the integral kernels coalesce. Generically, two saddle points coalesce, in which case the scaling function is expressible in terms of the Airy function. As we will see, this is the case for Dyck and Schröder paths, directed column-convex polygons and partially directed self-avoiding walks. The result for Dyck paths also allows for the scaling analysis of Bernoulli meanders (also known as ballot paths). We then construct the model of deformed Dyck paths, where three saddle points coalesce in the corresponding integral kernel, thereby leading to an asymptotic expression in terms of a bivariate, generalised Airy integral. dc.description.sponsorship Universität Erlangen-Nürnberg en_US Queen Mary Postgraduate Research Fund dc.language.iso en en_US dc.publisher Queen Mary University of London en_US dc.rights The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author dc.subject Mathematical Sciences en_US dc.subject combinatorial models en_US dc.subject two-dimensional lattice walks en_US dc.subject polygons en_US dc.title Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers en_US dc.type Thesis en_US
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