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Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models by Beaton, Nicholas R.; Owczarek, Aleksander L.; Xu, Ruijie is a scholarly article available to read on EtoBox.

What is Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models about?

Lattice paths in the quarter plane have led to a large and varied set of results in recent years. One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has involved a variety of techniques, some highly intricate and specialised. The famous Kreweras and reverse Kreweras walk models are two particularly interesting models, as they are among the only four cases which have algebraic generating functions. Here we investigate how the properties of the Kreweras and reverse Kreweras models change when boundary interactions are introduced. That is, we associate three real-valued weights $a,b,c$ with visits by the walks to the $x$-axis, the $y$-axis and the origin $(0,0)$ respectively. These models were partially solved in a recent paper by Beaton, Owczarek and Rechnitzer (2019). We apply the algebraic kernel method to completely solve these two models. We find that reverse Kreweras walks have an algebraic generating function for all $a,b,c$, regardless of whether the walks are restricted to end at the origin or on one of the axes, or may end anywhere at all. For Kreweras walks, the generating function for w

Author

Beaton, Nicholas R.; Owczarek, Aleksander L.; Xu, Ruijie

Published

2019

Language

EN