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Enumeration and asymptotics of random walks and maps

Friday 11th April 2008 - 15:30 to 16:15
INI Seminar Room 1

In this talk, I want to give a brief survey of what I did in analytic combinatorics (=using generating functions to enumerate combinatorial structures, and then using complex analysis to get the asymptotics).

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.

By full asymptotics, I mean an expansion like $$f_n \sim C A^n n^\alpha + C' A^n n^(\alpha-1) + C'' A^n n^(\alpha-2) + \dots$$ where $A$ is the growing rate and $\alpha$ the "critical exponent" of the corresponding combinatorial structure.

Namely, 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 an old work with Philippe Flajolet, Michèle Soria, and Gilles Schaeffer, and on work in progress with Bernhard Gittenberger.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons