National Research Network S9600
Analytic Combinatorics and Probabilistic Number Theory

Principal Investigator

Co-Investigator

Funded Researchers

Veronika Kraus
Johannes Morgenbesser

Description

The aim of this project is to describe asymptotic properties of several discrete objects, where the main emphasis is to provide a global picture. This will be done by approximating discrete objects (properly rescaled) by continuous limiting objects. The prototype of problems of this kind is the approximation of a random walk on the grid by a Brownian motion. Another example - that is closely related to the proposed project - is the approximation of the depth-first search of a rooted (Galton Watson) tree by a Brownian excursion. And many other examples are known in the literature.

In this project we will focus on three main themes, on the shape of trees, on functional equations, and on digital expansions. For all these topics we will use the concept of generating functions to solve the corresponding counting problems and then apply proper analytic and probabilistic tools in order to obtain asymptotic and distributional results.