Finite Groups of Random Walks and Periodic 4-bar Links
Published:Dec 26, 2025 10:46
•1 min read
•ArXiv
Analysis
This paper addresses two long-standing open problems: characterizing random walks in the quarter plane with finite groups and describing periodic Darboux transformations for 4-bar links. It provides a unified method to solve the random walk problem for all orders of the finite group, going beyond previous ad-hoc solutions. It also establishes a new connection between random walks and 4-bar links, completely solving the Darboux problem and introducing a novel concept of semi-periodicity.
Key Takeaways
- •Solves the Malyshev problem for random walks in the quarter plane with finite groups of any order.
- •Establishes a new relationship between diagonal random walks and 4-bar links.
- •Completely solves the Darboux problem for 4-bar links.
- •Introduces the concept of k-semi-periodicity for Darboux transformations.
Reference
“The paper solves the Malyshev problem of finding explicit conditions for random walks with finite groups and completely solves the Darboux problem for 4-bar links.”