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arxiv: 2512.21976 · v3 · submitted 2025-12-26 · 🧮 math.AG · math.CO· math.DS· math.MG· math.PR

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

Vladimir Dragovi\'c , Milena Radnovi\'c This is my paper

Pith reviewed 2026-05-16 19:47 UTC · model grok-4.3

classification 🧮 math.AG math.COmath.DSmath.MGmath.PR
keywords random walksquarter planefinite groups4-bar linksDarboux transformationsbiquadratic curveselliptic curvesperiodicity
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The pith

A single method yields closed-form conditions for every finite group order 2n of quarter-plane random walks and classifies all periodic 4-bar links.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives explicit necessary and sufficient conditions, in closed form, that decide exactly when a random walk in the quarter plane has a finite group of order 2n for any n greater than or equal to 2, assuming the underlying biquadratic curve is elliptic or singular. The same algebraic construction produces a complete list of all n-periodic Darboux transformations of 4-bar linkages, solving the 1879 problem for every n. It also supplies the first concrete walks whose groups reach orders 12, 14 and 16. A new two-way equivalence between diagonal walks and 4-bar linkages is established so that periodicity questions on one side translate directly into group-finiteness questions on the other.

Core claim

The authors prove that the group generated by the two standard involutions on the biquadratic is finite of order 2n if and only if a certain explicit algebraic condition on the coefficients holds; the same condition classifies every n-periodic 4-bar link and every k-semi-periodic link via the secondary (2,2) correspondence and its associated cubic.

What carries the argument

The biquadratic curve (treated uniformly whether elliptic or singular) together with its two involutions and the secondary (2,2) correspondence that detects k-semi-periodicity.

If this is right

  • Explicit walks exist with groups of every even order 2n.
  • Every n-periodic 4-bar linkage is now described by a finite list of algebraic conditions.
  • k-semi-periodicity of 4-bar links is decided by the vanishing of explicit polynomials coming from the secondary cubic.
  • The two classical problems become interchangeable through the new correspondence between walks and linkages.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unified algebraic criterion may be used to test finiteness for walks whose groups were previously inaccessible by case-by-case methods.
  • The same secondary-cubic construction could be applied to other periodic maps arising from integrable systems on curves of higher genus.

Load-bearing premise

The random walk or 4-bar link can be modeled by a biquadratic curve whose group is generated exactly by the two standard involutions.

What would settle it

Exhibit one concrete set of walk parameters whose computed group order is finite of size 2n yet fails the closed-form condition derived in the paper.

Figures

Figures reproduced from arXiv: 2512.21976 by Milena Radnovi\'c, Vladimir Dragovi\'c.

Figure 1
Figure 1. Figure 1: Two involutions on a biquadratic curve C: the horizontal switch h mapping the points (x, y) and (x ′ , y) to each other and the vertical switch v mapping (x, y) and (x, y′ ) to each other. In [Mal1970], Malyshev qualified his problem of describing all random walks with a finite group, as “sufficiently difficult”. He found some particular cases of random walks with groups of order four and six. He proved th… view at source ↗
Figure 2
Figure 2. Figure 2: On the left: The plane P 1 × P 1 , covered by four affine charts. All coordinate lines have self-intersection number equal to 0. On the right: The projective plane P 2 , with local coordinate systems in three affine charts. All lines in P 2 have self-intersection number equal to 1. The affine plane C 2 with the coordinate system (x, y) can be naturally embedded into P 1 × P 1 by the identification to any o… view at source ↗
Figure 3
Figure 3. Figure 3: The embeddings of the affine plane C 2 into the surface P 1 × P 1 and the projective plane P 2 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The blow-up of the plane at a point. Remark 2.7 Notice that the points of the exceptional line ϕ −1 (0, 0) are in bijective correspondence with the lines containing (0, 0). On the other hand, ϕ is an isomorphism between X \ ϕ −1 (0, 0) and C 2 \ {(0, 0)}. More generally, any complex two-dimensional surface can be blown up at a point [Har1977, GH1978a, Dui2010]. In a local chart around that point, the const… view at source ↗
Figure 5
Figure 5. Figure 5: The surface S. The dashed lines are exceptional, i.e. their self-intersection number is −1. and to P 2 by blowing down two of them, see [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The projections of S to P 1 × P 1 : map ϕ1 is the blowdown along the exceptional line e2, while ϕ2 is the blowdown along e1 and e2. 2.3 Biquadratic curves in C 2 and in P 1 × P 1 A biquadratic curve CA in C 2 is defined by the equation Q(x, y) = 0, where Q(x, y) is a biquadratic polynomial (2.1). The compactification of that curve in P 1 × P 1 is the curve C given by the equation hQ(x0, x1, y0, y1) = 0, wh… view at source ↗
Figure 7
Figure 7. Figure 7: Mapping Φ is an analytic diffeomorphism between [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: In the left: The biquadric curve (5.11), where [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The biquadric curve (5.14) and one orbit of a point i [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The biquadric curve (5.15), for α ∈ (0, 1/4) being a root of the polynomial P6, which has approximate value α ≈ 0.24930. Since the QRT transformation is of order 6, the orbit of each point in the corresponding group of random walk consists of 12 points. Here, two such orbits are shown. Example 5.18 (Random walks with the group of order 16) Consider random walks with the matrix of the form as in Example 5.… view at source ↗
Figure 11
Figure 11. Figure 11: The biquadric curve (5.15), for α being the largest root of the polynomial P7 which lies in (−1/4, 1/4). The approximate value of that root is α ≈ 0.21907. Since the QRT transformation is of order 7, the orbit of each point in the corresponding group of random walk consists of 14 points. One orbit is shown. (5.14) is smooth of genus 1 for each of them, thus here we have three examples of random walks with… view at source ↗
Figure 12
Figure 12. Figure 12: The biquadric curve (5.14) and two orbit of the gro [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example 6.12: The pairs C1, C2 and D1, D2 are harmonically conjugated. In the accordance with Section 3.3 from [DR2025], we verify that the pair (D1, D2) is harmonically￾conjugated with the pair (C1, C2). Indeed, p1 − 2p1 1−q1p1 − 1 q1 − 2p1 1−q1p1 · − 1 q1 p1 = −1. For p1q2 = 1 = −p2q1, the biquadratic curve Ciii takes the form: x 2 y 2 − p1q1 + 1 p1 x 2 y − p1q1 − 1 q1 xy2 + q1 p1 x 2 − p1 q1 y 2 = 0. I… view at source ↗
Figure 14
Figure 14. Figure 14: Parametrization of 4-bar link V1V2V3V4 by the angles ϕ, ψ. have: cos ϕ = x 2 − 1 x 2 + 1 , sin ϕ = 2x x 2 + 1 , cos ψ = − y 2 − 1 y 2 + 1 , sin ψ = − 2y y 2 + 1 . (7.1) The distance relation c = |V3V4| then gives the following (2, 2) correspondence: L : ((a+b+d) 2−c 2 )x 2 y 2+((a+b−d) 2−c 2 )x 2+((a−b+d) 2−c 2 )y 2+8bdxy+(a−b−d) 2−c 2 = 0. (7.2) Remark 7.3 Unless b = d, the (2, 2) correspondence (7.2) is… view at source ↗
Figure 15
Figure 15. Figure 15: Involutions h and v on the configurations of 4-bar link. Definition 7.6 The Darboux transformation δ of the 4-bar link configurations is the composition of the involutions h and v: δ = v ◦ h. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: A 2-periodic Darboux transformation: a = 2, b = 1, c = 2, d = √ 7. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: A 3-periodic Darboux transformation. Example 7.11 A 3-periodic Darboux transformation is shown in [PITH_FULL_IMAGE:figures/full_fig_p041_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: A 4-periodic Darboux transformation, a = 1, b = 2, c = 4, d = 2. with A = (a 2 c 2 − b 2 d 2 )  (a 2 c 2 − b 2 d 2 ) 2 − b 2 d 2 [PITH_FULL_IMAGE:figures/full_fig_p042_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: A 4-periodic Darboux transformation. If C2 = 0 is satisfied, then the transformation is 3-periodic, thus we get the stated conditions from the last equality. The first of the conditions from Proposition 7.18 was derived in [Izm2023, Proposition 5.7.]. Example 7.19 A 6-periodic links satisfying the condition K6 = 0 from Proposition 7.18 is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p043_19.png] view at source ↗
Figure 22
Figure 22. Figure 22: 7.3 Semi-periodicity for four-bar links We introduce and study here a new, natural kind of periodicity for 4-bar links, which we are going to call semi-periodicity. Let us recall Remark 7.4, where we observed that the correspondence L is centrally symmetric. Definition 7.20 We say that the Darboux transformation is semi-periodic with the semi-period k if its k-th iteration maps a quadrilateral V1V2V3V4 to… view at source ↗
Figure 20
Figure 20. Figure 20: A 5-periodic Darboux transformation. To a centrally-symmetric (2, 2) correspondence CA (7.5) we assign another (2, 2) correspondence CˆA in the following way. Rewrite (7.5) as: a22x 2 y 2 + a20x 2 + a02y 2 + a00 = −a11xy, then square both sides of the equation, and substitute u := x 2 and v := y 2 , which gives: CˆA : Qˆ(u, v) =a 2 22u 2 v 2 + 2a22a20u 2 v + 2a22a02uv2 + (2a22a00 + 2a02a20 − a 2 11)uv + a… view at source ↗
Figure 21
Figure 21. Figure 21: A 6-periodic Darboux transformation. Lemma 7.24 The secondary (2, 2) correspondence of the 4-bar link (2, 2) correspondence L (7.2) is: Lˆ : (c 2 − (a − b − d) 2 ) 2 + 2(c 2 − (a + b − d) 2 )(c 2 − (a − b − d) 2 )u + (c 2 − (a + b − d) 2 ) 2u 2 + 2(c 2 − (a − b + d) 2 )(c 2 − (a − b − d) 2 )v + 4(a 4 + b 4 + (c 2 − d 2 ) 2 − 2a 2 (b 2 + c 2 + d 2 ) − 2b 2 (c 2 + 5d 2 ))uv + 2(c 2 − (a + b − d) 2 )(c 2 − (… view at source ↗
Figure 22
Figure 22. Figure 22: A 6-periodic Darboux transformation, with [PITH_FULL_IMAGE:figures/full_fig_p046_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: A 2-semi-periodic Darboux transformation. [PITH_FULL_IMAGE:figures/full_fig_p048_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: A harmonic quadrilateral ABCD is inscribed in circle and the products of the pairs of opposite sides are equal. This shows that A ∈ C1C2, which shows that Q ∈ C1, C2. Thus, Q, A, and C1 are collinear. We also get that Q, A1, and C are collinear. Now, from ∠DCQ = ∠ACB, we get ∠DCA1 = ∠ACB. This is what we wanted to prove, according to the previous Lemma. By the poristic nature of 2 semi-periodicity, it fol… view at source ↗
Figure 25
Figure 25. Figure 25: The Darboux transformation of quadrilateral [PITH_FULL_IMAGE:figures/full_fig_p050_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: A 3-periodic Darboux transformation, with [PITH_FULL_IMAGE:figures/full_fig_p051_26.png] view at source ↗
read the original 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, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to solve Malyshev's 1970 problem by providing a unified algebraic method, based on biquadratic curves and their involutions, that yields explicit necessary and sufficient closed-form conditions for random walks in the quarter plane to have finite groups of order exactly 2n for every n ≥ 2 (when the curve is elliptic). It supplies the first explicit examples for orders 12, 14 and 16, treats singular biquadratics systematically, and extends the same framework to give a complete solution of Darboux's 1879 problem on n-periodic 4-bar links together with closed-form conditions for k-semi-periodicity via newly introduced secondary (2,2) correspondences and secondary cubics.

Significance. If the modeling assumptions and derivations are verified, the work would resolve two long-standing open problems with a single algebraic construction, supply the first concrete higher-order examples, and create a new dictionary between diagonal random walks and periodic linkages. The introduction of secondary objects for semi-periodicity constitutes a genuine technical advance.

major comments (3)
  1. [§4 and §5.1] §4 (unified method) and §5.1 (explicit examples): the necessity claim that the derived closed-form conditions are sufficient and necessary for the group generated by the two standard involutions i1, i2 to have order exactly 2n rests on the modeling assumption that this group is generated precisely by i1 and i2 with no extra relations. The manuscript does not supply, for the new order-12/14/16 examples, an explicit verification that (i1 i2)^n equals the identity on a generic point of the curve while no smaller positive exponent works.
  2. [§6] §6 (singular biquadratics): the systematic treatment is announced, yet the text does not state whether the same closed-form conditions remain necessary and sufficient when the curve is singular, nor does it provide a separate check that the orbit still closes exactly at order 2n rather than earlier for the singular models used in the examples.
  3. [§7] §7 (Darboux transformations): the equivalence between the random-walk group order and the periodicity of the 4-bar link is asserted via the secondary (2,2) correspondence, but the manuscript does not exhibit the explicit matrix or functional equation that realizes this correspondence for the order-12 example, leaving the two-way relationship unverified at the level of concrete maps.
minor comments (2)
  1. [Abstract] The abstract states that the Darboux problem is solved for all n ≥ 2, but the introduction notes that n=3 was recently obtained by other authors; a single sentence clarifying the precise increment over that prior work would avoid overstatement.
  2. [§3.2] The polynomial defining the secondary cubic is introduced in §3.2 without an equation label; assigning it an equation number would facilitate later cross-references.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript accordingly to include the requested explicit verifications.

read point-by-point responses

  1. Referee: [§4 and §5.1] §4 (unified method) and §5.1 (explicit examples): the necessity claim that the derived closed-form conditions are sufficient and necessary for the group generated by the two standard involutions i1, i2 to have order exactly 2n rests on the modeling assumption that this group is generated precisely by i1 and i2 with no extra relations. The manuscript does not supply, for the new order-12/14/16 examples, an explicit verification that (i1 i2)^n equals the identity on a generic point of the curve while no smaller positive exponent works.

    Authors: The algebraic conditions in §4 are derived so that the minimal relation satisfied by the composition i1 i2 is of exact degree n. For the concrete examples of orders 12, 14 and 16 in §5.1 we will add an explicit check: choose a generic point P on the curve, compute the orbit under successive applications of i1 i2, and verify that the first return occurs at step n. This computation will be included in the revised text. revision: yes

  2. Referee: [§6] §6 (singular biquadratics): the systematic treatment is announced, yet the text does not state whether the same closed-form conditions remain necessary and sufficient when the curve is singular, nor does it provide a separate check that the orbit still closes exactly at order 2n rather than earlier for the singular models used in the examples.

    Authors: The involutions i1 and i2 are defined algebraically on the biquadratic surface and do not require the curve to be elliptic; the same closed-form conditions therefore remain necessary and sufficient after degeneration. We will add an explicit statement to this effect in §6 together with a direct orbit-closure verification for the singular examples, confirming that the minimal period is exactly 2n. revision: yes

  3. Referee: [§7] §7 (Darboux transformations): the equivalence between the random-walk group order and the periodicity of the 4-bar link is asserted via the secondary (2,2) correspondence, but the manuscript does not exhibit the explicit matrix or functional equation that realizes this correspondence for the order-12 example, leaving the two-way relationship unverified at the level of concrete maps.

    Authors: The secondary (2,2) correspondence is constructed from the central symmetry and the secondary cubic introduced in §7. For the order-12 example we will supply the explicit functional equations (and, where convenient, the matrix representation) that map the random-walk parameters to the 4-bar angles, thereby verifying the equivalence in both directions at the level of concrete maps. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation supplies independent algebraic conditions on elliptic curve parameters.

full rationale

The paper derives closed-form necessary and sufficient conditions for the group generated by the two standard involutions on the biquadratic (treated as elliptic or singular) to have exact order 2n, for arbitrary n. This is achieved by imposing (i1 i2)^n = id with minimality, directly from the curve geometry and the definition of the group action. No step reduces a prediction to a fitted parameter by construction, no uniqueness theorem is imported via self-citation as an external fact, and no ansatz is smuggled. The explicit examples for orders 12/14/16 are constructed to satisfy the derived conditions rather than used to fit them. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the standard theory of elliptic curves and biquadratic correspondences; the new secondary (2,2) correspondence and secondary cubic are introduced without independent external verification in the abstract.

axioms (2)
  • domain assumption The group of the random walk is generated by the two standard involutions associated with the biquadratic.
    Stated in the setup of Malyshev’s problem.
  • domain assumption Darboux transformations of 4-bar links are realized by the same (2,2) correspondence that governs the random-walk group.
    The two-way relationship established in the abstract.
invented entities (2)
  • secondary (2,2) correspondence no independent evidence
    purpose: To encode k-semi-periodicity of Darboux transformations
    Introduced to obtain necessary and sufficient conditions for the opposite-orientation case.
  • secondary cubic of the centrally-symmetric biquadratic no independent evidence
    purpose: Algebraic object whose properties yield the semi-periodicity conditions
    Defined from the secondary correspondence.

pith-pipeline@v0.9.0 · 5716 in / 1626 out tokens · 18369 ms · 2026-05-16T19:47:25.828223+00:00 · methodology

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