Enumeration and Asymptotics of Random Walks and Maps

Abstract

In this talk, I want to give a brief survey of what I did in analytic combinatorics (generating functions and complex analysis). 　　　This survey will be based on 3 kinds of equations which are often met in combinatorics, the way we solve them, and what kind of generic methods we use to get the full asymptotics/limit laws. 　　　I will show that three combinatorial structures are "exactly solvable": - a directed random walk model (using the kernel method and singularity analysis of algebraic functions), - random walks on the honeycomb Lattice (using an Ansatz and Frobenius method for D-finite functions), - question of connectivity in planar maps (using Lagrange inversion and coalescing saddle points, leading to a ubiquitous distribution involving the Airy function). 　　　This talk is based on different works, some of them with Philippe Flajolet, Michele Soria, and Gilles Schaeffer, or more recently with Bernhard Gittenberger.