Some reflections on directed lattice paths

TL;DR

This paper studies directed lattice paths with boundary conditions, analyzing how reflection and absorption at the boundary affect generating functions, asymptotics, and limit laws, extending previous models with more complex analytic behavior.

Contribution

It introduces a detailed analysis of reflected lattice paths, deriving asymptotics and limit laws, and highlights the impact of boundary conditions on the combinatorial and probabilistic properties.

Findings

Generated functions are modified by boundary conditions.

Asymptotics for lkiewicz walks are provided.

Limit laws for returns and final altitude are established.

Abstract

This article analyzes directed lattice paths, when a boundary reflecting or absorbing condition is added to the classical models. The lattice paths are characterized by two time-independent sets of rules (also called steps) which have a privileged direction of increase and are therefore essentially one-dimensional objects. Depending on the spatial coordinate, one of the two sets of rules applies, namely one for altitude 0 and one for altitude bigger than 0. The abscissa y=0 thus acts as a border which either absorbs or reflects steps. The absorption model corresponds to the model analyzed by Banderier and Flajolet ("Analytic combinatorics of directed lattice paths"), while the reflecting model leads to a more complicated situation. We show how the generating functions are then modified: the kernel method strikes again but here it unfortunately does not give a nice product formula. This makes the analysis more challenging, and, in the case of {\L}ukasiewicz walks, we give the asymptotics for the number of excursions, arches and meanders. Limit laws for the number of returns to 0 of excursions are given. We also compute the limit laws of the final altitude of meanders. The full analytic situation is more complicated than the Banderier-Flajolet model (partly because new "critical compositions" appear, forcing us to introduce new key quantities, like the drift at 0), and we quantify to what extent the global drift, and the drift at 0 play a role in the "universal" behavior of such walks.