Walks in the quadrant with interacting boundaries : genus zero case

TL;DR

This paper classifies the generating functions of genus zero lattice walks with interacting boundaries, revealing they are mostly hypertranscendental, with some algebraic cases depending on boundary contact weights.

Contribution

It extends the classification of lattice walk generating functions to models with boundary interactions, using $q$-difference equations and analyzing algebraic relations among weights.

Findings

Most generating functions are hypertranscendental.

Certain weight relations lead to algebraic or rational generating functions.

Complete classification for all real boundary interaction weights in genus zero models.

Abstract

The study of lattice walks restricted to the first quadrant has shed a lot of interest in the past twenty years. In particular, there has been an important effort to classify models of weighted walks with small steps with respect to the algebraic-differential nature of their generating function. The techniques that were developed in the course of this work are now applied to different extensions of those walks. One of these extensions, called walks with interacting boundaries, consists in accounting for the number of contacts of the walk with the axes, with motivation coming from statistical physics. These contacts are encoded as two additional parameters for the generating function, the Boltzmann weights. For one notable family of models, called genus zero models, we establish in this paper the complete classification of their generating function, for all real values of the parameters. We do this by adapting to this more general case a method due to Dreyfus, Hardouin, Roques and Singer, used in the former classification, and which consists in studying the rational solutions to a $q$-difference equation. In almost all cases, we show that the generating function is hypertranscendental, regardless of the values of the weights. In the remaining cases, we prove that specific algebraic relations between the Boltzmann weights make the generating function $\mathbb{N}$-algebraic or $\mathbb{N}$-rational, contrasting with the interaction-less case.