Finite Groups of Random Walks in the Quarter Plane and Periodic $4$-bar Links

TL;DR

The paper resolves Malyshev's long‑standing problem by formulating a unified algebraic‑geometric framework based on biquadratic curves and QRT transformations, linking finite‑order random‑walk groups in the quarter plane to finite‑order translations on an associated elliptic cubic. It provides explicit necessary and sufficient conditions for group orders $4,6,8,10$ (and higher) using determinant/coefficient criteria and associated invariants, including singular cases, with closed‑form expressions. A novel two‑way correspondence between diagonal quarter‑plane random walks and planar four‑bar links is developed, connecting discrete probability, elliptic geometry, and Darboux transformations, and yielding a complete periodicity/semi‑periodicity theory for 4‑bar links. The results have implications for the algebraicity of generating functions in lattice‑path combinatorics and furnish new tools to study discrete integrable systems across probability, geometry, and mechanics.

Abstract

We solve two long standing open problems, one from probability theory formulated by Malyshev in 1970 and another one from a crossroad of geometry and dynamics, going back to Darboux in 1879. The Malyshev problem is of finding effective, explicit necessary and sufficient conditions in the closed form to characterize all random walks in the quarter plane with a finite group of the random walk of order $2n$, for all $n\ge 2$, in the generic case where the underlining biquadratic is an elliptic curve. Until now, the results were known only for $n=2, 3, 4$ and were obtained using ad-hoc methods developed separately for each of the three cases. We provide a method that solves the problem for all $n$ and in a unified way. We also consider situations with singular biquadratics. Further, we establish a new two-way relationship between \emph{diagonal} random walks in the quarter plane and $4$-bar links. We describe all $n$-periodic Darboux transformations for $4$-bar link problems for all $n\ge 2$, thus completely solving the Darboux problem, that he solved for $n=2$. We introduce \emph{$k$ semi-periodicity} as a novel and natural type of periodicity of the Darboux transformations, where after $k$ iterations of the Darboux transformation, a polygonal configuration maps to a congruent one, but of opposite orientation. By introducing new objects, \emph{the secondary $(2-2)$-correspondence} and the related \emph{secondary cubic} of the centrally-symmetric biquadratics, we provide necessary and sufficient conditions for $k$-semi-periodicity for $4$-bar links for all $k\ge 2$ in an explicit closed form.

Figures (17)

• Figure 1: The plane $\mathbb{P}^1\times\mathbb{P}^1$. All coordinate lines have self-intersection number equal to $0$.
• Figure 2: The projective plane, with local coordinate systems in three affine charts. All lines in the plane have self-intersection number equal to 1.
• Figure 3: The blow-up of the plane at a point.
• Figure 4: The surface $\mathcal{S}$. The dashed lines are exceptional, i.e. they have the self-intersection number equal to $-1$.
• Figure 5: Parametrization of $4$-bar link $V_1V_2V_3V_4$ by the angles $\varphi$, $\psi$.

Theorems & Definitions (81)

• Definition 2.1
• Proposition 2.2
• Theorem 2.3: Frobenius, Fr
• Corollary 2.4
• Theorem 2.5
• Definition 2.6
• Proposition 2.7: Cayley, Cayley1871
• Definition 3.1
• Remark 3.2
• Proposition 3.3