Enumeration and asymptotics of random walks and maps
Seminar Room 1, Newton Institute
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.
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