Infinite Singular Systems and Random Discrete Objects

Discrete objects appear in connection with computers and algorithms almost every day, for example as data structures or as networks. In recent decades, various methods have developed in mathematics to analyze discrete objects. The first mathematical question is usually how many different objects of a certain size there are, i.e. a combinatorial counting problem. Another question is what properties a discrete object typically has. Formally, one equips the objects with a probability distribution and looks for statements that occur with high probability, e.g., what distance can be expected in large networks. A very powerful mathematical method to answer these two questions are so-called generating functions, i.e. power series, whose coefficients encode the number of objects or also the probability distributions. The structure of the discrete objects is usually reflected in mathematical properties such as functional equations or even systems of functional equations for the generating functions. An exact mathematical analysis of such functional equations or systems of functional equations then leads to solutions of the addressed questions. In particular, in many cases, even if an exact solution is not achievable, an approximate solution can be obtained which is asymptotically as meaningful as an exact solution. The goal of the project is to transfer asymptotic results for finite systems of functional equations to infinite systems. Infinite systems do indeed occur very often, but there are very few mathematical results so far. These shall be investigated systematically, first for so-called reducible systems, then for systems with an additional catalytic variable, and finally for almost-periodic linear and almost-periodic nonlinear systems.