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Universality classes for weighted lattice paths: where probability and ACSV meet

Banff International Research Station for Mathematical Innovation and Discovery

Lattice paths are very classic objects in both probability theory and enumerative combinatorics. In particular, weighted models bridge the gap between the two approaches very neatly. We consider an example, the Gouyou-Beauchamps model of lattice walks in the first quadrant, and discuss how to determine asymptotic enumeration formulas parameterized by the weights. The major tool is the theory of analytic combinatorics in several variables (ACSV) and we identify six different kinds of asymptotic regimes (called universality classes) which arise according to the values of the weights. Because we are able to explicitly and generically compute the constants of the asymptotic formula, we can determine a formula for a family of discrete harmonic functions. Furthermore, we are able to demonstrate an infinite class of models for which the counting generating function is not D-finite.

Mathematics; Combinatorics; Probability theory and stochastic processes; Control/optimization/operation research

video/mp4

Author affiliation: Mathematics, Simon Fraser University

2017-04-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61393

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/