Work

The Weighted Walks in Quadrant with Finite Groups: an Algebro-Geometric Approach

Year: 2026

Type: preprint

Abstract: We classify weighted small-step lattice walks in the quadrant whose associated birational group is finite. Using an algebro-geometric description of the kernel curves and Cayley-type finite-order conditions, we relate the group $G_W$ of a walk to the family of groups $Γ_t$ acting on its kernel curves. Together with the uniform upper bound on the order of $G_W$, this allows us to analyse all possible finite orders. We obtain explicit necessary and sufficient conditions for the group to have order 4, 6, 8, or 10, and prove that no weighted walk in the quadrant has a group of order 12. This yields a complete classification of weighted quadrant walks with finite groups.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.04657

Authors: Dragovic, Vladimir, Radnovic, Milena

Institutions: The University of Texas at Dallas, Mathematical Institute of the Serbian Academy of Sciences and Arts, The University of Sydney

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7211870352

Cites: 3

Cited by:

Related to: 10