CanaDAM 2021
On-line, May 25 - 28, 2021

Generating series and confined lattice walks - Part II
Org: Thomas Dreyfus (CNRS, Université de Strasbourg) and Andrew Elvey Price (CNRS, Université de Tours)

ALIN BOSTAN, INRIA Saclay Île-de-France
On the D-transcendence of generating functions for singular walks in the quarter plane  [PDF]

In their recent article “Walks in the quarter plane: Genus zero case”, Thomas Dreyfus, Charlotte Hardouin, Julien Roques and Michael F. Singer used Galois theory of difference equations to study the nature of the generating function $Q(x,y,t)$ of walks in the quarter plane for the so-called "singular models", i.e., with kernel curve of genus zero. They proved that for transcendental values of $t$, the generating function is differentially transcendental both in $x$ and in $y$. I will discuss the situation when one also looks at algebraic values of $t$. This is joint work with Lucia Di Vizio and Kilian Raschel.

MANUEL KAUERS, Johannes Kepler Universität
Quadrant Walks Starting Outside the Quadrant  [PDF]

We investigate a functional equation which resembles the functional equation for the generating function of a quarter plane lattice walk model. It has the interesting feature that its orbit sum is zero while its solution is not algebraic. The solution can be interpreted as the generating function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject to the restriction that the coordinate axes can be crossed only in one direction. We also consider certain variants of the equation, all of which seem to have transcendental solutions. This is joint work with Manfred Buchacher and Am\'elie Trotignon.

IRÈNE MARKOVICI, Université de Lorraine
Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude  [PDF]

I will present some connections between two families of walks. The first family is formed by two-dimensional walks moving in three directions, and confined within a triangle. The other family consists of Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. After showing a symmetry property for the triangular paths, I will describe different bijections between these two families of walks, answering an open question of Mortimer and Prellberg.

This is a joint work with Julien Courtiel and Andrew Elvey Price.

MARNI MISHNA, Simon Fraser University
Lattice Walk Classification: algebraic, analytic, and geometric perspectives  [PDF]

This talk will examine the rich topic of lattice path enumeration. Recent attention on classifiction has brought together techniques from many mathematical sub-disciplines. In this talk, we will see how lattice walks arise in algebraic combinatorics, and illustrate the results of enumerative techniques that use analysis and geometry. One goal of this talk is to clarify the connection between algebraic sources of lattice walks and certain classes of generating functions, especially D-finite. A second outcome is a geometric understanding of the asymptotic enumeration formulas for weighted models.