Limit laws for discrete excursions and meanders and linear functional equations with a catalytic variable

We study limit distributions for random variables defined in terms of coefficients of a power series which is determined by a certain linear functional equation. Our technique combines the method of moments with the kernel method of algebraic combinatorics. As limiting distributions the area distributions of the Brownian excursion and meander occur. As combinatorial applications we compute the area laws for discrete excursions and meanders with an arbitrary finite set of steps and the area distribution of column convex polyominoes. As a by-product of our approach we find the joint distribution of area and final altitude for meanders with an arbitrary step set, and for unconstrained Bernoulli walks (and hence for Brownian Motion) the joint distribution of signed areas and final altitude. We give these distributions in terms of their moments.Comment: 33 pages, 1 figur

Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers

PhDThe subject of this thesis is the asymptotic behaviour of generating functions 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.Universität Erlangen-Nürnberg Queen Mary Postgraduate Research Fun

FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONS

International audienc

Nanoinformatics

Machine learning; Big data; Atomic resolution characterization; First-principles calculations; Nanomaterials synthesi

Nanoinformatics

Machine learning; Big data; Atomic resolution characterization; First-principles calculations; Nanomaterials synthesi

Research and technology annual report, 1982

Various research and technology activities are described. Highlights of these accomplishments indicate varied and highly productive reseach efforts