Finite Groups of Random Walks in the Quarter Plane and Periodic $4$-bar Links
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, of Darboux from 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 the finite group of random walk of order $2n$, for all $n\ge 2$, where the underlining biquadratic is an elliptic curve. Until now, the results were known only for $n=2, 3, 4$, 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. Explicit examples of random walks with the groups of orders higher than 10 are presented here for the first time, including orders 12, 14, 16. The same method applies to any higher order. We consider cases with singular biquadratics in a systematic manner. We establish a new two-way relationship between diagonal random walks and $4$-bar links. We describe all $n$-periodic Darboux transformations for $4$-bar links for all $n\ge 2$, thus completely solving the Darboux problem: after $n$ iterations, a polygonal configuration maps to a congruent one of the same orientation, that he solved for $n=2$, which was recently extended to $n=3$. We also study $k$-semi-periodicity as a 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 a new object, the secondary $(2,2)$ correspondence, and the related 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, while the case $k=2$ was solved recently.