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

  • Vladimir Dragovic
  • Milena Radnovic
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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 GW of a walk to the family of groups Γt acting on its kernel curves. Together with the uniform upper bound on the order of GW, 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.

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Cite

BibTeXRIS
BibTeX
@misc{dragovic2026weighted,
  title = {{The Weighted Walks in Quadrant with Finite Groups: an Algebro-Geometric Approach}},
  author = {Vladimir Dragovic and Milena Radnovic},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04657},
  eprint = {2609.04657},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04657},
}
RIS
TY  - UNPB
TI  - The Weighted Walks in Quadrant with Finite Groups: an Algebro-Geometric Approach
AU  - Vladimir Dragovic
AU  - Milena Radnovic
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04657
UR  - https://arxiv.org/abs/2609.04657
AB  - 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.
ER  -