SFU Discrete Math Seminar: Marni Mishna

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. Work in collaboration with Julien Courtiel, Stephen Melczer, and Kilian Raschel.